-
EXERCISE 9.1 BASIC INTEGRATION FORMULAS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 1
1. 62 4 + 5
= 63
3
42
2+ 5 +
= + +
3. ( 1)
=
= 3
2
=
+
5. 22+43
2
= 2 + 4
3
2
= 2 + 4
3
2
= 2 + 4
31
1
= + +
+
7. 38
2
= Factor, (x-c), c = 2 P(c) = 0 the (x-c ) is the factor P(c) =
0 2 1 0 0 -8 2 4 8 1 2 4 0
= (2+2+4)(2)
(2)
= (2 + 2 + 4)
= 3
3+
2
2
2+ 4 +
=
+ + +
9. 4 23 + 2
= 2 22
3 +
=
+
+
-
EXERCISE 9.2 INTEGRATION BY SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 2
1. 2 3
Let u = 2 - 3x
= 3
3=
= 1
2(
3)
= 1
3
1
2
= 1
3
232
3 +
=
+
3. 2(23 1)4
Let u = 23 1
= 62
6= 2
= 2(23 1)4
= (4)(
6)
=1
6 (4)
=1
6 5
5 +
=5
30+
=( )
+
5. (2+3)
2+3+4
Let u = 2 + 3 + 4
= 2 + 3
= (2 + 3)
=
= +
= + + +
7. 2
(31)4
Let u = 3 1
= 32
3= 2
=
3
4
= 1
3 4
= 1
3 3
3 +
= 3
9+
=
()+
-
EXERCISE 9.2 INTEGRATION BY SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 3
9.
2
Let u =
=
1
=
= 1
2(
)
= 1
2
= 2
= 1
1+
=
+
11.
1
Let u = =
= (1
1
1
)
= 1
1
1
Let v = 1
=
= 1
1
= +
= 1 ; =
= ln| 1| ln| | +
= ln(1 )
= +
13. cos4 sin
Let u = cos
= sin
= sin
= 4
= - 4
= 5
5+
=
+
15. 1 + 2 sin 3 3
Let u = 3
= 3
3=
= 1 + 2 sin (
3)
= 1
3 1 + 2 sin
Let v =1 + 2 sin
= 2 ;
2=
= 1 + 2 sin 3 3
= 1
3[
1
2 (
2)]
= 1
6
232
3 +
= (+)
+
-
EXERCISE 9.2 INTEGRATION BY SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 4
17. 2
+ `
Let u = +
=
sin
;
= 2
=
= 1
=
+ +
19. 3 23
Let u = 3
= 323 ;
3= 23
= 1
2(
3)
= 1
3[
232
3] +
= 1
3
2tan 332
3 +
= ()
+
21. 32+14+14
+4
= ()
() = +
()
* using synthetic division
-4 3 14 13
-12 -8
3 2 5 - R(x)
= 3 + 2
+ 4 = ()
= (3 + 2) + 5
+4
For the second integral :
= + 4 ;
= 1 ; =
= (3 + 2) + 5
= [32
2+ 2 + 5 + ]
=
+ + ( + ) +
-
EXERCISE 9.2 INTEGRATION BY SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 5
23. 5232
2+1
3 3
2 + 1 5 23 2
5 + 3
33 2
33 3
()
()dx = +
()
()
= 3 3 +
2+1
=4
4
32
2+
2+1
For the 2nd term
Let u = x2+1
= 2
2=
=4
4
32
2+
2
=
+
+ +
-
EXERCISE 9.3 INTEGRATION OF TRIGONOMETRIC FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 6
1. 55
= 5
= 5
5=
=
5
= 1
5
= 1
5 +
=
+
3. +
2
=
2 +
2
= 1
+
= +
= + +
5.
sin1
2 cot
1
2
; Let u= 1
2
=
1
2 2 =
= 2
= 2
= 2
(
)
= 2 1
()
= 2
= + +
7. cos 3
1.
1+
1+
= (cos 3 ) 1+
(1 )(1+ )
= (co s3 +cos 3 )
1sin 2
= cos 3 1+
cos 3
= 1 +
= +
=
= ; =
= +
= +2
2+
= +
+
9. 1 + 2
= (1 + 2 + tan2 )
= [2 + (1 + tan2 )]
= 2 + sec2
= 2 || + +
= || + +
-
EXERCISE 9.3 INTEGRATION OF TRIGONOMETRIC FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 7
11. 6
cos 2 3
Let u = 3x ; 2u = 6x
= 3 ;
3=
= 2
3
cos 2
= 2
3
1
= 2
3
=
+ +
13. 2
2 2
= 2
(2 )
=
= 1
=
= + +
15. 4 sin 2 2
2 2
= (4 )( )
2 2
= 2
2
= 2
2
Let u = 2x
= 2
2=
.
2
= 1
2
=
+
17.
3 3
Let u = 3x
= 3
3=
=
3
=1
3 +
=
+
-
EXERCISE 9.4 INTEGRATION OF EXPONENTIAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 8
1.
2
= 2dx
= 2 ;
= 2 ;
2=
= (
2)
= 1
2
=1
2 +
=1
2+
=
() +
3. 44
= 4 ;
= 4 ;
4=
= (
4)
= 1
4
= ;
= cos ; =
=1
4
= 1
4 +
=
+
5. 3 = 3
2
= 3
2 ;
2
3=
= (2
3)
= 2
3
= 2
3
3
2 +
=
+
7. 532
= 3 2 ;
= 2 ;
2=
= 5 (
2)
= 1
2 5
= 1
2
532
5 +
=
+
9. 32
= ()
= 6
=
+
-
EXERCISE 9.5 INTEGRATION OF HYPERBOLIC FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 9
1. 3 1
Let u = 3 1
= 3 ; =
3
= (
3)
= 1
3
= 1
3 + c
=
+
3. 2 1 2 Let u=1 2
= -2
2=
= 2(
2)
=1
2 2
=1
2( + )
=
+
5. 2
= ;
=
1
; =
= 2
= +
= () +
7. 1
2
1
2
Let u = 1
2 ;
=
1
2 ; 2 =
= 2
= 2( + )
=
+
-
EXERCISE 9.6 APPLICATION OF INDEFINITE INTEGRATION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 10
1. Given slope 32 + 4
= 32 + 4
= 32 + 4
= 32 + 4
=33
3+ 4 +
= + +
3. Given slope +1
1
=
+ 1
1
1 = + 1
2 2 =2
2+ + 2
2 2 = 2 + 2 + 2
+ + + =
5. Given slope 1
=
1
=
2
2=
ln 2
2+ 2
= +
7. Given slope 2
, through 1,4
=
2
2=
1
4= +
ln 1
4=
ln 1 1
4=
= 1
4
ln 1
+
1
4= 0 4
4 ln 4 + = 0
+ =
9. Given slope , through 1,1
=
1
2 =
1
2
1
2
= +
21
2 = +
When = 1 , = 1
2 1 = 1 + ; = 1
21
2 = + 2
= +
-
EXERCISE 9.6 APPLICATION OF INDEFINITE INTEGRATION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 11
11. Given slope 2, through 1,2
=
1
2
=
2
= 1
+
2 = 1
1+
2 = 1 +
= 3
= 1
+ 3 x
= 1 + 3
+ =
13.
a=-32 ft/sec2
a=-2
= 32
= 32
v=-32t+c
= 32 + 1
= (32 + 1)
s=16t2 + c1t + c2
when t = 0, v = vo
v=-32t + c1
vo= -32(0) + c1
vo =c1
v = -32t + vo
when t = 1 sec, s=h=48ft
h=-16t2+ vot + c1
48 = -16(1)2 + vo(1) + c2
64 - vo = c2
When t = 0, s = 0, c2 = 0
s = -16t2 + vot
when t = 1 sec, s = 48
s = -16t2 + c1t
48 = -16(1)2 + c1(1)
c1=64
s=-16t2 + 64t
v = -32t + 64
@ max, v = 0
0 = -32t + 64
32t=64
t = 2 sec
s = -16t2 + 64t
s = -16(2)2 + 64(2)
s = 64ft
-
EXERCISE 9.6 APPLICATION OF INDEFINITE INTEGRATION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 12
15.
a = 32ft/sec2
a = 32
= 32
= 32
v = 32t + c1
= 32 + 1
= 32 + 1
S = 16t2 + c1 + c2
when t = 0, v = 0
c1 = 0
v = 32t
when t = 0 , s = 0
c2 = 0
s = 16t2
= 400
16
= 20
4
t = 5 sec
v = vt
*since it is a free falling body, its velocity is ( - )
vt = -32t
vt = -32(5)
vt = -160 ft/sec
-
EXERCISE 10.1 PRODUCT OF SINES AND COSINES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 13
1. sin 5 sin
= 2 sin sin
= [cos cos( + )]
= 5 =
=1
2[cos 5 cos(5 + )]
=1
2[cos 4 cos 6]
=1
2[ cos 4 cos 6
=1
2[
1
4sin 4
1
6sin 6 ] +
=
+
3. sin 9 3 cos + 5
=1
2 [sin 9x 3 + x + 5 + sin 9 3 5
=1
2[sin 5 + 2 + sin(3 8)]
= 5 + 2 ; = 3 8
= 5 ;
= 3
5= ;
3=
=1
2[ cos
1
5
1
3] +
=
+
+
5. cos 3 2 cos +
=1
2[cos + + cos( )]
= 3 2
= +
+ = 3 2 + +
= 4
= 3 2 +
= 2 3
=1
2[cos 4 + cos(2 3)]
cos 4 = cos 4 + 4
= 4
cos 2 3 = cos 2 3 + sin 2 3
= cos 2
=1
2(cos 4 cos 2)
=1
2[
1
4sin 4
1
2sin 2] +
=
+
-
EXERCISE 10.1 PRODUCT OF SINES AND COSINES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 14
7. 4 8 3
= 2[sin 8 + 3 + 8 3
= 2[11 + sin 5]
= 11 ; = 5
= 11 ;
= 5
11= ;
5=
= 2[1
11cos 11
1
5cos 5 ] +
=
+
9. 5 4 +
3 2
6
=5
2[cos cos( + )]
= 4 +
3 ;
= 4 +
3 2
6
= 2 + /2
= 2
6
+ = 4 +
3 + 2
6
= 6 + /6
=5
2[ 2 +
2 6 +
6 ]
cos 2 +
2
= cos 2
2 sin 2 sin
2
= 2
cos 6 +
6
= 6
6 6
6
= 3
2 6
1
2 6
=5
2[ 2
3
2 6 +
1
2 6 ]
=5
2[
1
2 2
3
12 6
1
12 6 +
=
+
-
EXERCISE 10.2 POWER OF SINES AND COSINES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 15
1. 3 4;
= 44
= (1 2 )24
= (1 22 + 4 )4
= (4 26 + 8 )
Let u = cosx
=
- =
= - (4 26 + 8)
= 26
7
5
5
9
9 +
=
+
3. 4 333 ;
= 432 33
= 4 3 1 23 3
= (4 3 63)3
Let u = sin3x
= 33 ;
3= 3
= ( 4 6)
3
= 1
3 5
5
7
7 +
= 1
155
1
217 +
=
+
5. 4 2
= (2 )22
= (1 2
2)2
1 + 2
2
= 1 22 +
1
4 (22
4
1 + 2
2
= 1
4 (
1
22 +
1
422)
1
2+
2
2
= 1
8 1 22 + 22 1 + 2
= 1
8 (1 22 + 2 + 2 22 2 + 32)
= 1
8 (1 2 2 2 + 32)
= 1
8 2 2 2 + 3 2
= 1
8 [
1
22 (
1
2 +
1
84 +
1
22
1
632]
=
+
-
EXERCISE 10.2 POWER OF SINES AND COSINES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 16
7. ( + )2dx
= ( + 2 + 2)
= + 2 1
2 + 2
= + 2 1
2 + (1+2
2)
Let u = sinx
=
=
= + 2 1
2 + 1
2 +
2
2
= - + 2 2
3
3
2 +
2+
2
4+
= - +
+
+
+
9. (3 + 2)2
= 23 + 232 + 22
= 23 + 2 32 + 22
=
2
1
26
1
5 5 +
2+
1
84 +
=
+
+
11. 2 4
= 1 + 8
2
=1
2 1 + 8
=
+
+
. 3 2
= 2 2 2
= 1 2 2 2
= 2 ; = 22
= 1 2
2
=1
2
3
3 +
=
+
+
. 7 2
= 7 2
= 7 1 2
= 7 9 u=sinx du=cosxdx
= 7 9
=8
8
10
10+
=
+
-
EXERCISE 10.3 POWER OF TANGENTS AND SECANTS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 17
1. 2242
= 222222
= 22(1 + 22)22
= (22 + 42)22
= 2 ;
= 222
2= 22
= (2 + 4)(
2)
= 1
2 (2 + 4)
= 1
2
3
3+
5
5 +
=
+
+
3. 6 ;
= 1
242
= 1
2(1 + 2)22
= 1
2(1 + 22 + 4)2
= (1
2 + 25
2 + 9
2)2
= ;
= 2 ; = 2
= (1
2 + 25
2 + 9
2)
= 2
32
3+
472
7+
2 112
11 +
=
+
+
+
5. ____1
2 . =
2
3
3
2 2
2+ +
:
= 2
22
2 2
2+ 1
= 2
22
2 (2
2 1)
= 2
22
2 2
2
= 2
2(2
2 1)
= 2
2(2
2)
= 4
2dx
= , ""
7. ( + )2
= (2 + 2 + 2)
= + 2 + 2
= + 2 + (2 1)
= + 2 + +
= + +
-
EXERCISE 10.3 POWER OF TANGENTS AND SECANTS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 18
9. (3
3)4
= 43
43
= 4343
= 232343
= 23(1 + 23)43
= (43 + 23)23
= 3 ;
= 32 3
3= 23
=1
3 (4 + 2)
= 1
3 3
3
2
3 +
= 1
3 33
3
13
3 +
=
+
11. 3
= 3
1
2
= 2
3
2
= (2 1)
3
2
= (1
2 3
2)
= ;
=
=
= (1
2 3
2)
= 2
32
3 2
1
2 +
=
+
-
EXERCISE 10.4 POWER OF COTANGENTS AND COSECANTS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 19
1. 44
= 4(1 + 2)2
= (4 + 6)2
= ;
= 2 ; = 2
= (4 + 6)
= - 5
5+
7
7+ c
= -
+
+ c
3. 54
= 3424
= 34(24 1)
= (3424 34)
= [3424 (24 1)4]
= 3424 424 4
= 4 ;
4= 24
=1
4 3
1
4 +
1
4(4)
=1
4
4
4
2
2 +
1
4 4 +
=
+
+
+
5. 3 4 3
= 1
2 3 2 3 2 3
= 1
2 3 1 + 2 3 2 3
= 1
2 3 + 5
2 3 2 3
= 3
= 3 2 3
3= 2 3
= 1
3
1
2 + 5
2
= 1
3
32
3
2
+
72
7
2
+
=
+
-
EXERCISE 10.4 POWER OF COTANGENTS AND COSECANTS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 20
7. 5 2
8 2=
5 2
5 2
1
3 2
= 5 2 3 2 \
= 4 2 2 2 2 2
= 2 2 1 2 2 2 2 2
= 4 2 2 2 2 + 1 2 2 2 2
= 6 2 2 4 2 + 2 2 2 2
= 2
= 2 2 2
()
2= 2 2
= 1
2 6 24 + 2
= 1
2
7
7
25
5+
3
3 +
=
+
+
9. 4
6
= 6 2 2
= 6 1 + 2 2
= 6 + 4 2
: =
= 2
= 2
= 1 6 + 4
= 1 5
5
3
3 +
= 5
5+
3
3+
=
+
+
-
EXERCISE 10.5 TRIGONOMETRIC SUBSTITUTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 21
1. 2
42
= ; = 2 ; 22 2
=
=
2
= 2
= 2
= 2
4 2
= 4 2
2
= 4 2
= 4 1 2
2
= 2 1
2 2 + ; = (
2)
= 2[
2
1
2 2 ]2 + C
=
+
3.
92+4;
= 3
= 2
=
3 = 2
=2
3 ; =
3
2
=2
32
= 92 + 4
2
2 = 92 + 4
=
92 + 4
=
2
32
2
32
=
2
= 1
2
1
= 1
2
1
= 1
2
= 1
2[-| + |] +
=
+
+
-
EXERCISE 10.5 TRIGONOMETRIC SUBSTITUTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 22
5. 2
92 32
= 2
9 2 3
= 2
9 2 9 2
= ; = 3
=
= 3
= 3
= (3)2 3
9 (3)2 3
= 92
1 2
= 2
(1 2)
=2
2
= 2 2
=
=
+
7. 942
2
= 3 ; = 2
= ; 2 = 3
=3
2
2
3=
=3
2
= 1(2
3)
= 3 (
3
2 )
(3
2 )2
= 3 (3 )
2(3
2 )2
= 9 2
2(9
4 2)
-
EXERCISE 10.5 TRIGONOMETRIC SUBSTITUTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 23
9.
2+4 2 ; : = , = 2
= 2 ; =
2
= 2 2 ; =
2
2 + 4 x
2
= 2 + 4
2
2 = 2 + 4 2
4 2 = 2 + 4
= 2 2
4 2 2
= 2 2
16 4 =
8 2 =
1
8
2
= 1
8 2
=1
8
1 + 2
2
=1
8
1
2 +
2
2
=1
8
1
2 +
1
4(2) +
=
+
+
11.
29
= 3 ; =
=
= 3 ; = 3
=
3 ; =
3
= 2 9
3 ; 3 = 2 9
=
2 9 =
3
3(3)
=
3
= 1
3
=
+
2 9
x
3
-
EXERCISE 10.5 TRIGONOMETRIC SUBSTITUTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 24
. ( 2 16
3
2
3)
= ; = 4
= ; = 4 ; =
4
=
4
3 = 64 sec3 ; = 4
= 2 16
4; 4 = 2 16
= ( 4 3(4))
(64sec3)
= 4 tan4
sec^ 2
= 4 (sec2 1)^2
sec2
= 4 sec4 2 sec2 + 1
sec2
= 4 sec4 2 sec2 + 1
sec2
= 4 sec2 2 + 1/ sec2
= 4( 2 +1
2 +
=
+
+
.
2 3 5 12 + 42
5 12 + 42 = 2 9 4
= 2 ; = 2 3
= ; 2 3 = 2
2 3 = 2 ; 2 = 2 + 3
2 = 2
=
=2 3
2
= 2 3
2
= 2 3 2 4
2
2 = 2 3 2 4
= ()
22
=1
4
=1
4
=1
4; =
23
2
=
+
-
EXERCISE 10.6 ADDITIONAL STANDARD FORMULAS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 25
.
2 + 25
Let: =
= 5
=
=
+
.
1 4
Let: = 2
= 1
2=
=1
2
2
1+
=
+
.
49 252
Let: = 7
= 5
2=
=
+ +
. 36 92
Let: = 6
= 3
3=
=1
3 3
2 36 92 +
1
3 8
3
6 +
=1
3
3
2 36 92 +
1
3 8
2 +
=
+
+
. 162 + 25
Let: = 5
= 4
4=
=1
4
4
2 162 + 25 +
1
4
52
2 4 + 162 + 25 +
=
+ +
+ + +
-
EXERCISE 10.7 INTEGRANDS INVOLVING QUADRATIC EQUATIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 26
1.
23+2
2 3 = 2
2 3 =9
4= 2 +
9
4
3
2
2
=1
4
3
2
2
1
4
=
( 3
2)2
1
4
= 3
2
=1
2
=
2 2=
1
2
+ +
=1
2 1
2
3
2
1
2
3
2+
1
2
+
=
+
.
22 2 + 1
=
1
2
2+
1
4
=
2 + 2
=1
2
+
=1
2, =
1
2
=1
2
1
21
2
+
=
+
. 3 2 2
= 4 + 1 2
= + 1, = 2
= 2 2 =
2 2 +
2
2
+
=+1
2 3 2 2 +
4
2
+1
2+
= +
+
+
+
-
EXERCISE 10.7 INTEGRANDS INVOLVING QUADRATIC EQUATIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 27
.
2 8 + 7
Completing the square
2 8 = 7
2 8 + 16 = 7 + 16
4 2 = 9
4 2 9 = 0
=
( 4)2 + 9
= 3 ; = 4
=
2 2
=1
2
+ +
=1
6
43
4+3 +
=
+
9. 3+2
2+9
= 3
2+9+
2
2+9
= 3
2+9+ 2
2+9
= 2 + 9 ; = 2
= 31
3
3+ 2
2
=
+ + +
11. 23
421
= 2
421
3
421
= 2
421 3
421
= 42 1 ;
8=
= 2
8
3[
1
2
421
42+1 + ]
=
| |
+ +
13. (2+7)
2+2+5
= 2+2 +5
2+2+5
= 2+2
2+2+5+ 5
(+1)2+4
= 2 + 2 + 5 ; = (22)
=
+
1
2
+1
2+
= | + + | +
+
+
-
EXERCISE 10.7 INTEGRANDS INVOLVING QUADRATIC EQUATIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 28
15. (3)
42
= 2 1
42
= 2
42
42
= 4 2 ; =4 2
2 4 2
= 2(2)
2 42 ; 4 2 = 4 (2 )2
=
4(2)2
=
+
17. +3
82
= 4 +7
82
= 4
82+ 7
82
= 8 2 ; =8 2
2 8 2
=2( 4)
2 8 2; 8 2
= 16 (4 )2
= + 7
16(4)2
= - +
+
19. (4+9)
24+20 =
2(2+4+17)
24+20
= 2 2+4
24+20+
17
2
24+20
= 2 4 + 20 ; = (2 4)
2 4 + 20 = 2 2 + 16
= 2[
+
17
2
2 2+16]
= 2[ 2 4 + 20 +17
2(
1
4)Arctan
2
4+ ]
= + +
Arctan
+
-
EXERCISE 10.8 ALGEBRAIC SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 29
5
1.
23
= 3
= 3 2
3 2
= 3
1
= 1
=
= 3
= 3 +
= 3 | 1 | +
= |
| +
3. (
13
14
412
= 12
= 1211
= 3 (43) 11
8
= 3 (9 8)
= 3[ 9 8]
= 3[10
10
9
9+ ]
=310
10
9
3+
=3
5
6
10
7
4
3+
=9
5610
34
30+
=
(
)
+
5.
+2 34 +2
12
= + 24
4 = + 2
= 2 4
= 43
= 4 3
3 2
= 4
1
= 1
=
= + 1
= 4 + 1
= 4[
+
]
= 4[ + +
= [ + + ]
-
EXERCISE 10.8 ALGEBRAIC SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 30
7. 4 + ;
=(4+ )1/2
2 4 = 4 82 + 16 =
= 12
(2 4)2= = 43 16
= 43 16
= 44 162
= 4 5
5 16
3
3 +
= 4
5(4 + )5/2
16
3(4 + )3/2+C
= 4 + 3
2 4
5 4 +
16
3 +
= 4 + 3
2 12 4 + 80
15 +
= 4 + 3
2 48 + 12 80
15 +
= 4
15 4 +
3
2 12 + 3 20 +
=
+
+
9. + 4 1
3
= + 4 1
3 ; 3 = + 4
= 3 4 ; = 33
= 3 4 32
= 3 6 4 3
=3
7
3
7 3
4
3 +
= 3 + 4
7
3
7 3 + 4
4
3 +
=3 +4
43
7 + 4 7 +
+ 4 1
3 = +
+
-
EXERCISE 10.8 ALGEBRAIC SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 31
11. 4 2+1
12
= 2 + 1 ; 2 = 2 + 1
2 = 2 1 ; =2 1
2 ; =
= 4
1 2 21
2
= 4 2
2 2
= 1 +4 2
2 2
= 1 4 2
2 2
= 2 2 1
2 2
= 2 2
2 2
2 2
= 2 2 2 +1
2
21 2
2+1+ 2 +
= + +
+ + +
. x5 4 + x3 dx
= 4 + 32 = 4 + 3 ; = 4 23
=1
3 4 2
2
3 2
= 2
3 4 2 2
3
= 5 4 + 3
= 4 23
5
() 2
3 4 2 2
3
= 4 2 2
3
= 82 + 24
3
=1
3 24 82
=1
3 25
5
83
3 +
=25
15
83
9+
=6 4 + 3 40 4 3
45+
= 4 + 3 6 4 + 3 40
45+
= 4 + 3 24 + 63 40
45+
= 4+3 16+63
45+
= +
+
-
EXERCISE 10.8 ALGEBRAIC SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 32
15. 3(4 + 2)3
2
3 + 1
2= 2
Z = 4 + 2
2 = 4 + 2
X = 4 2
dx = 1
2 4 2
1
2(-2zdz)
=-
(42)12
= (4 2)(3)()
= 44 + 6
= 45
5+
7
7+
=285 + 57
35+
=28( 4+2 )5+5( 4+2)7
35+C
= 4 + 2
5(28 + 5(4 + 2)
35+
= 4+2
5(28+20+52)
35+
= +
( )
+
17. 1
4 2+1
= ; = 2
2 + 1 = 2 + 1 = &
= 1
= 3
= 2( 1)
= . = =
= 2 1
= 1 2
= 3
3
= 3
3+
= +
+
-
EXERCISE 10.8 ALGEBRAIC SUBSTITUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 33
. (
2(81 + 4)
=1
; =
1
2
2
1
2 81 +
1
4
2
814+1
6
= 3
814 + 1 3
4
= 814 + 1 ; = 3243
=1
324
3
4
=1
81 814 + 1
1
4 +
=
+
+
21. (3)1/3
4
=1
2, =
1
=
1
1
3
2
1
4
=
21
3
1
3
2
1/4
=
(21)
1
3 (
2)
4
= 2 1 1
3
= 2 1
= 1
2
1
3
= 1
2(
43
4
3
)+c
= 3
8 2 1
4
3 +
= 3
8
1
2 1
4
3+
= 3
8
1-x2
x2
4
3+c
RyanRectangle
-
EXERCISE 10.9 INTEGRATION OF RATIONAL FUNCTIONS OF SINES AND
COSINES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 34
1.
1+
= 2 1+2
1+12
1+2
= 2 1+2
1+2+ 12
1+2
= 2
2
= = +
=
+
3.
4+2 =
2
1+ 2
4+2 2
1+2
= 21+2
4+ 4
1+2
= 21+2
4+42+ 4
1+2
= 2
42 + 4 + 4 ; : = +
1
2 ; =
3
2
= 2
+ 1
2
2+
3
4
= 2
2 + 2=
2
+
=2
3
2
+
1
2
3
2
+ =1
3
2+1
3+
=
+
+
.
+ + 3=
21+2
2
1+2+
12
1+2+3
= 21+2
1+222
1+2+ 3
= 21+2
1+222+3+32
1+2
= 2
4 + 2 + 22=
2
+1
2
2+
7
4
= 2
2 + 2 =
7
2, = +
1
2
=2
+ =
2
7
2
+
1
2
7
2
+
=1
7
2+1
7+
=
+
+
. = 1 + 2
1 2 .
2
1 + 2
= 2
1 2= 2
2 2 = 1, =
=2
2
+
+ =
1+
1 +
=
+
+ =
+
+
-
EXERCISE 10.10 INTEGRATION BY PARTS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 35
1. = ; = =
= =
=
= + +
3. 2 ; = 2 ; = ; = 2 ; =
= 2 ; = - ; = 22 ; = -
= -2 22
= -2 2 2
= -2 2[-2 -22
= -2 + 22 4 2
4 2
= 222
5+
=
+
5. 2 ; = ; = 2
= ; =2
1+2
= 2 2
1+42
= 2 2
1+42
= 2 1
4
=
+ +
-
EXERCISE 10.10 INTEGRATION BY PARTS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 36
7. 3 ; = 2 ; =
= =
= 2
= 2 1
= + 3 ; 3
2 3 = + +
3 =
+ + +
9. 22 ; = 2 ; =
= 1
2 +
1
84 =
= 1
2 +
1
84 (
1
2 +
1
84)
=2
2+
1
84
1
42 +
1
324 +
=
+
+
+
11.
12 ; =
12 ; =
= - 1 2 ; =
12
= - 1 2 (- 1 2)(
12)
= - 1 2 +
= +
-
EXERCISE 10.10 INTEGRATION BY PARTS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 37
13. 1 + ; = 1 + ; =
= 1
1+ ; =
= - 1 + + 2
1+
= - |1 + | + 1 2
1+
= - 1 + + 1
= - 1 + + + +
= + + + +
15.
(+1)2 ; = ; =
1
(+1)2
= + 1 ; = - 1
+1
= -
+1+ =
++
17. 2; = ; =
12 ; = 2 ; =
3
3
= 1
3
1
3
3
12
= 1
33 + (
1
3
3
9) +
= 1
33 +
3 12
9 +
= 1
33 +
12 2+2
9+
; 1
3
3
12 ; = 1 ; = ; = ; = ; 1 2 =
=1
3
3
()
=1
3 2
=
( +
) +
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 38
1. 12+18
+2 +4 (1)
12 + 18
+ 2 + 4 ( 1)=
( + 2)+
( + 4)+
( 1)
12 + 18 = + 4 1 + + 2 1 + + 2 ( + 4)
12 + 18 = (2 + 3 4) + 2 + 2 + (2 + 6 + 8)
12 + 18 = 2 + 3 4 + 2 + 2 + 2 + 6 + 8
2 + 2 + 2 = 0
3 + + 6 = 12
4 + + 8 = 18
= 1
= 3
= 2
=
(+2)+
3
(+4)+
2
(1)
= + + +
3.
1 (4)
1 =
( 1)+
( 4)
1 = 4 + ( 1)
1 = 4 +
+ = 0
4 = 1
=
=1
3
= 1
3
(1)
+ 13
(4)
= 1
3 1 +
1
3 4 + =
+C
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 39
5. 62+239
(3+223)
62 + 23 9
+ 3 ( 2)
62 + 23 9 =
+
( + 3)+
( 1)
62 + 23 9 = + 3 1 + 1 + ( + 3)
62 + 23 9 = 2 + 2 3 + 2 + (2 + 3)
+ + = 6
2 + 3 = 23
3 + 0 + 0 = 9
= 3
= 2
= 5
= 3
2
(+3)+ 5
(1)
= + + +
7. 3+52+9+7
2+5+4
3 + 52 + 9 + 7
+ 4 ( + 1)
By division of polynomials,
5 + 7
+ 4 ( + 1)=
( + 4)+
( + 1)
5 + 7 = + 1 + ( + 4)
= 4,
=13
3
= 1
=2
3
= +
13
3
( + 4)+
2
3
( + 1)
=
+
+ +
+ +
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 40
9. 2+1
2 (3)2
2 + 1 =
( 2)+
( 3)+
( 3)2
2 + 1 = 3 2 + 3 2 + 2
2 + 1 = 2 6 + 9 + 2 5 + 6 + 2
+ = 0
6 5 + = 2
9 + 6 2 = 1
= 5
= 5
= 7
= 5
2 +
5
3 +
7
3 2
= 5 2 5 3 +7
3
=
+
( )
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 41
11. 25
(1)
2 5
( 1)=
+
( 1)+
( 1)2+
( 13
2 5 = 1 3 + 1 2 + 1 +
2 5 = 3 32 + 3 + 3 22 + +C 2 +
2 5 = 3 = 32 22 + + 2 +
2 5 = 3 + 2 32 22 + 2 + 3 + +
2 5 = + 3 + 3 2 + 2 + 3 + +
+ = 0
3 2 + = 0
3 + + = 2
= 5
= 5
= 5
= 5
= 3
= 5
+
5
( 1)+
5
( 1)2+
3
( 1)3
= 5
5
( 1)+ 5
( 1)2 3
( 1)3
= 5 5 1 5
(1)+
3
2(1)2+
=
( )+
( )+
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 42
13. 32+17+32
3+82+16
32 + 17 + 32
( + 4)2
32 + 17 + 32
( + 4)2=
+
( + 4)+
( + 4)2
+ = 3
8 + 4 + = 17
16 = 32
= 2
= 1
= 3
= 2
+
( + 4)+
3
( + 4)2
= + + +
+
15. 2+1
31 (2+2+2)
2 + 1
3 1 (2 + 2 + 2)=
(3 1)+
2 + 2 +
2 + 2 + 2
2 + 1 = 2 + 2 + 2 + 2 + 2 3 1 + 3 1
2 + 1 = 2 + 2 + 2 + (62 + 4 + 2) + 3 1
+ = 0
2 + 4 + 3 = 2
2 + 2 = 1
= 5
2
=5
2
= 1
= 5
2
(3 1)+
5
2
(2 + 2)
2 + 2 + 2
2 + 2 + 2
= 5
2 3 1 +
5
2 2 + 2 + 2 2 + 2 + 2
=
+ +
+ +
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 43
17. 52+17
+2 (2+9)
52 + 17
+ 2 (2 + 9)=
+ 2+
2 +
2 + 9
52 + 17 = 2 + 9 + 2 + ( + 2)
52 + 17 = 2 + 9 + 22 + 4 + + 2
52 + 17 = 2 + 22 + 4 + + 9 + 2
52 + 17 = + 2 2 + 4 + + 9 + 2
2 = + 2 = 5
= 4 + = 1
= 9 + 2 = 17
+ 2 = 5 2 = 2 4 = 10
4 + = 1 =4 + = 1
2 + = 11
2 + = 1 2 = 4 2 = 22
9 + 2 = 17 =9 + 2 = 17
13 = 39
A=3
9(3)+2C=17 4B-5=-1
27+2C=17 4B=-1+5
2C=17-27 4B=4
2C=-10 B=1
C=-5
= 3
+ 2+
1 2 5
2 + 9
=3
+2+ 2
2+9 5
2+9
= + + +
+
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 44
19. 42+21+54
2+6+13
4 3 2
2 + 6 + 13
2 + 6 +
2 + 6 + 13
2 + 6 + = 3 2
2 + = 3
= 11
=3
2
= 4 [3
2
2 + 6
2 + 6 + 13+ (11
2+ 6 + 13)]
= 11
2 + 6 + 9 + 13 9
= 11
+ 3 2 + 13 9 2
= 11(1
2
+3
2)
= 4 3
2| 2 + 6 + 13|
11
2
+3
2
=
| + + | +
+
+
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 45
21. 3+72+25+35
2+5+6
+ 2 +9 + 23
2 + 5 + 6
9 + 23
+ 3 ( + 2)=
+ 3+
+ 2
9 + 23 = + 2 + ( + 3)
x=-3
9(-3)+23= A(-3+2)+B(-3+3)
-27+23=A(-1)+B(0)
-4=-A
A=4
If x=-2
9(-2)+23= A(-2+2)+B(-2+3)
-18+23=A(0)+B
5=B
B=5
= + 2 +2
+ 3+
5
+ 2
= + 2 4
+3+ 5
+2
=
+ + + + +
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 46
23. 28
(23)(2+2+2)
2 3 +
2 2 +
2 + 2 + 2
A(2 + 2 + 2) + 2 + 2 2 3 + (2 3)
A(2 + 2 + 2) + 42 2 6 + (2 3)
A+4B=1
2A-2B+2C=-1
2A-6B-3C=-8
A=-1
2
A=1
2
C=1
1
(2 3)+
1
2
2 + 2
2 + 2 + 2+
2 + 2 + 2
(2 3) +1
22 + 2 + 2 +
+ 1 2 + 12
= 1
2 2 3 + 2 + 2 + 2 + + 1 +
=
+ +
+ + +
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 47
25. 5+233
2+1 3
= 5 + 23 3
6 + 34 + 22 + 1
= 2 +
2 + 1+
2 +
2 + 1 2+
2 +
2 + 1 3 2 + 1 3
= 2 2 + 1 2 + 2 + 1 2 + 2 2 + 1 + 2 + 1 + 2 +
= 2 4 + 22 + 1 + 4 + 22 + 1 + 23 + 2 + 2 + 1 + 2 +
= 25 + 43 + 2 + 4 + 22 + 1 + 23 + 2 + 2 + 1 + 2 +
5: 2 = 1 ; =1
2
4: = 0 ; = 0
3: 4 + 2 = 2 ; = 0
2: 2 + = 0 ; = 0
: 2 + 2 + 2 = 3 ; = 0
: + + = 0 ; = 0
=
+ +
+ +
-
EXERCISE 10.11 INTEGRATION OF RATIONAL FUNCTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 48
27.
4 + 23 + 112 + 8 + 16
(2 + 4)2
[
+
2 +
(2 + 4)+
2 +
(2 + 4)2][(2 + 4)2]
A 2 + 4 2 + 2 (2 + 4) + 2 + 4 () + (2)() + ()
A(4 + 82 + 16) + 24 + 82 + 3 + 4 + 22 +
4: + 2 = 1 A = 1
3: = 2 B = 0
2: 8A+8B+2D=11 C = 2
X: 4C + E=8 D = 3/2
C : 16A = 16 E = 0
=
+
2
2 + 4+
3
2
2
(2 + 4)2
= + 2 1
2
2
3
2 2+4 +
= +
+ +
-
EXERCISE 11.1 SUMMATION NOTATION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 49
= 10
. 123
=1
= 12 3
=10
=1
= 12 102 10 + 1 2
4
= 3(100 121 )
=
. (122 + 4
=10
=1
)
= 12 2 + 4
=10
=1
=10
=1
= 12 10(10 + 1)(2 10 + 1)
6 + 4
10(10 + 1)
2
= 2 110 21 + 2 110
=
. ( 1)( + 1)
=10
=1
= 3
=1
= 3
=10
=1
= 3 +
=10
=1
=10
=1
=102 10+1 2
4
10 10+1
2
=
. + =
=
= 92=10
=1
+ 6 + 1
= 9 2 + 6 + 1
= 9 10(10+1)(2 10 +1)
6 + 6
10(10+1)
2 + 10
=
. + + + ( )
=
=
-
EXERCISE 11.1 SUMMATION NOTATION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 50
. 1 1 + 2 2 + +
= ()
=
. 14 + 24 + 34 + + 4
=
=
. 11+22+33 + +
=
=
. 13 + 2
3 + 33+ +
3
=
=
-
EXERCISE 11.2 THE DEFINITE INTEGRAL
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 51
1 0 0 0 1
1 0
. 322
1
= 0 ; = 2
= 2 0
=2
= +
= 0 + 2
=2
= =
2
3
=1
(2
)
= 3
42
2 (
2
)
= 3
82
3
=
24 ( + 1)(2 + 1)
6
1
3
= 24 2+1 2+1
6 3
= 24 23+2+22+
63
=
3. 2 ( 1)1
0
= 0 ; = 1
=1 0
; =
= 2
2
= {
2 [ ( 2
2)
1
]
3 }
= 2 1
2
+1 2+
6
1
2
2
=
2 1
3
23 + 2 + 22 +
6 1
=2
3 1
=
5. 2 + 3 5
1
=5 1
; = 1 +
4
=4
== (1 +4
)
4
+ 3
== 4
+
16
2+ 3
==4
+
16
2+
(+1)
2 + 3
=
-
EXERCISE 11.2 THE DEFINITE INTEGRAL
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 52
. 32
0
=2
; =
2
=
2
3
2
=
83
3
2
=
16
4 2 + 1 2
4 3
=
4
4(2(2 + 2 + 1)
= 44
4+
8
3+
42
3
= 4 + 0 + 0
= 4
-
EXERCISE 11.3 SOME PROPERTIES OF THE DEFINITE INTEGRAL
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 53
. 32 2 + 12
1
= 33
3+
22
2+
= 8 4 + 2 1 + 1 1
= 5
. 32 +4
2
3
1
= 33
3+
4
= 27 4
3 1 + 4
=
. 1 + 23
7
0
= 1 + 2
= 2
=1
2
3 1+2 43
4
=
.
2 + 1
3
2
= 2 + 1
= 2
=1
2
3
2
= 1
2 10 5
= .
9.
(2+21)
0
1
=
( + 2 + 1 1 1)
0
1
=
[ + 1 2 + 2]
0
1
=
+ 1 2 + 2
0
1
=
2 + 1 2
0
1
= 2 ; = ( + 1)
= +1
2 +
=
-
EXERCISE 11.3 SOME PROPERTIES OF THE DEFINITE INTEGRAL
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 54
.
2 +
0
= 2 + ; = 2 ;
2=
=
2
=1
2
=1
2
0
=1
2 ln 2 +
0
=1
2 ln 2 + 0 + ; =
=1
2
2 +
=
1
2
+ 1
=1
2 + 1 = + 1
1
2
= +
.
2+4
2
0 u= x; du=dx; a=2
= 1
2
2
= 1
2 1
=
. 2 21
0
2 1
0
cos = 2
2 ; 2 = 2
=
2 ; 2 =
= 22 ; = 4 = 1, = 4 ; = 0, = 0
= 2 2
4
4
= 8 22
4
0
= 8 1 2
2
1 + 2
2
4
0
= 2 1 22
4
0
=
. 1
0
= ; =
= ; =
= 1
0=
= 1 1 + 0 1 = 1
-
EXERCISE 11.3 SOME PROPERTIES OF THE DEFINITE INTEGRAL
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 55
. 2
2
0
= ; =
= 2
2
= 3
3
= 3
3
=
. 6 4
2
= 61 63 65 (41)(43)(
2
)
(6+4)(6+42)(6+44)(6+46)(6+48)
=
. 7
2
=(41)(73)(75)
7(72)(74)(76)
=
. 6
2
0
2
2
=
2 ; =
2
= 2 6 2
= 2 61 63 65 21
6+2 6+22 6+24 6+26
2
=
. 2 4 2 2
4
8
= 2 1 2 1
2 + 2 2 + 2 2
=1
4 2
2
=
. 4 2 3
2
2
0
; = 2
= 2
= 4 2 2 3
2
2
0
(2)
= (4 2 )3
2
2
0
2
= 8 3 2 2
0
= ( 41 43
4 42
2
=
-
EXERCISE 12.1 AREA UNDER A CURVE
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 56
2
1
2
1
1. = 32 ; = 1 = 2
= 2
1
= 322
1
= 3
= 2 3 1 3
= .
3. = 1 ; = 1 = 2
= 1
= 2
1
= 1
2
1
= [ ]
= {[ 2] [ 1]}
= 2; ,
,
= .
-
EXERCISE 12.1 AREA UNDER A CURVE
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 57
3
0
5. = 3, = 2 = 4
=
0
= 3 4
2
= 3[ ]
= 3[4 4 4] 3[2 2 2]
= 3[4 4 4 2 2 + 2]
= 3[82 22 2]
= 3[62 2]
= 6[32 1]
= [ ] .
7. = 9 2 ; = 3 = 3
= 4 2 3
3
=
9. + = 3 &
= 3 3
0
= 3 3
2
= 3 3 3 2
2
=
.
-
EXERCISE 12.1 AREA UNDER A CURVE
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 58
11. 2 = 4, = 1 = 4
= 44
1
= 41
24
1
= 8
3
3
4
=8(4)3/2
3
8(1)3/2
3
=64
3
8
3
=
.
. = 1, = , = 2, = 0
= 1; =
() = 1
= 1 ; = 1 ; (1,1)
1 = 1
2
1
= ( )
= 2 1
1 = 2 .
2 =1
2
=1
2 1 1
2 = 1
2.
= 1 + 2
= ( +
).
-
EXERCISE 12.2 AREA BETWEEN TWO CURVES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 59
3 -1
2
-1
1. = 2 ; = 2 + 3
= 2 + 3
2 = 2 + 3
2 2 3 = 0
3 + 1 = 0
= 3, = 1
= 2 + 3 2 3
1
= [2 + 3 3
3]
= 32 + 3(3) (3)3
3 (1)2 + 3(1)
(1)3
3
= 9 +5
3
=
.
3. 2 = 1 ; = 3
Y1=Y2
3 2 = 1
2 6 + 9 = 1 5 2 = 0
= 5 , = 2
= 5 3 = 2
= + 3 (2 + 1) 2
1
= + 2 2 2
1
= 2
2+ 2
3
3
= 22
2+ 2(2)
23
3
1 2
2+ 2(1)
(1)3
3
=10
3+
7
6 = A =
27
6 =
.
5. y = x2 ; y = 2 x2
= 2 ; (0,0)
= 0 , = 0
2
2= 2 ( )
:
y1= y2
2 = 2 2
2 2 + 2 = 0
(2 + 2)( 1)
2 + 2 = 0 1 = 0
2 = 2
2 = 1
= 1 = 1
= [1 2]
= (2 2 2)1
1
= (2 22)1
1
= 2 23
3 = 2
2
3 [2 +
2
3]
= 2 2
3+ 2
2
3 =
124
3
=
.
RyanRectangle
RyanRectangle
-
EXERCISE 12.2 AREA BETWEEN TWO CURVES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 60
7. = ; = ; =
4 =
2
2 =
2
4
= [-]
= [-
4] [-
2] =
2
2
1 =
2
4
= =
2
4
= 1 2
2
= .
9. 2 = 4 , =8
2+4
=2
4
2 2 + 4 = 32
= 8
2 + 4
2
4
2
2
= 4.95
11. = 3 , = 8, = 0
= 32 , 0 = 32
= 0 , = 0
2
2= 6( )
:
y1= y2
3 = 8
3 8 = 0
3 = 8
= 83
= 2
= 2
= 8 , (2,8)
= 2
= 2 3 , = 8
(-2,-8)
= [1 2]
= (8 3)2
0
= = .
x y
0 0
90 1
180 0
270 -1
360 0
x y
0 1
90 0
180 -1
270 0
360 1
RyanRectangle
RyanRectangle
RyanRectangle
-
EXERCISE 12.2 AREA BETWEEN TWO CURVES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 61
13. = 2 + 1 , = 7 , = 8
= 2 + 1 7 8
2
= 2 + 1 7 + 8
2
= 3 6 8
2
= 32
2 6
= 3(8)2
2 6(8)
3(2)2
2 6(2)
=
15. = 3 , = ; =
= (2 1
1
)
= [(3
1
) ()]
= 3
1
1
= 3 ; = ; = ; =
= 32
3 = ; =
; =
= 3 (32
3
1
)
= 3 3
1 [ (
)]
1
= 3 3 1 [ ]
1
= .
8
2
-
EXERCISE 12.2 AREA BETWEEN TWO CURVES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 62
a
-a
17. 2 = 2 , 2 = 4 2
2 = 22 = 4 2
= 2
2 ; x =
2+2
4
=
2
2
=
0 = 0 ; (0,0)
2
2=
1
:
X1 = X2
2
2=
2 + 2
4
42 = 22 + 23
42 22 23 = 0
22 23 = 0
22 = 23
2 =23
2
2 = 2
= 2
=
X1 = X2=2
2=
2
= 4
=(4)2
2
=162
2
= 8
= 4
= (4)2
=162
2
= 8
= [2 + 2
4
2
2]
= (2+22
4
)
= 3
12+
2
4
23
12
= 3
12+
2
4
23
12
()3
12+
2()
4
2()3
12
= 3 23 + 3 23
12+
3 + 3
4
=23
12+
23
4 =
23+63
12
=43
12
A= a2
3sq. units
RyanRectangle
-
EXERCISE 12.2 AREA BETWEEN TWO CURVES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 63
. 2 = + 1 ; = 1
1=2 1; = 1
= 2 ; 2 = 1
= 0; = 0
2
2= 2 ( )
1=2 ; 2 1 =
1
2 + 2 = 0
( 1)( + 2)
1 = 0 + 2 = 0
y=1 y=-2
= 0 = 3
= 1, = 2
= 2, = 5
= 1, = 0
= 2, = 3
= 3, = 8
;
= 2 1
= 1 2 1 1
2
1
2
= 1 2 + 1 21
= 2 2 21
= 2 2
2
3
3 2
1
= 2(1) (1)2
2
(1)3
3 = 2(2)
(2)2
2
(2)3
3
= 2 1
2
1
3+ 4 + 2
8
3
=
.
. 2 = 4 ; = 4 4
4 = 22 = + 4
= 2
4 =
+ 4
2
=
1
42
0 =1
42
0 = 0
0,0
2
2 = (concave to the right)
2
4=
+ 4
2
22 4 + 4(4)
22 4 16 = 0
2 8 + 2
2 8 = 0 + 2 = 0
= 4; = 4(1, 2)
(4, 4)
= (2 1)
= +4
2
2
4
4
2
= .
RyanRectangle
-
EXERCISE 12.2 AREA BETWEEN TWO CURVES
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 64
23. 2 = + 4 , 2 + 1 = 0
= 2 1 2 4 3
1
= 3 + 2 2 3
1
= 3 + 2 3
3 1
3
=
25. = 2 , = , = 2
= 2 2
0
= 2
2
0
2
=4
2 2
1
2+ 1
=
-
EXERCISE 12.4 VOLUME OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 65
1 2
-1 y
dx (1,-1)
-2
1. = 2 2 , ,
= 2 2 ,
0 = 2 2 ; = 12 2(1)
= 1 ; = 1
2
2= 2
1, 1
x 0 1 2 3
y 0 -1 0 3
= 2
= 2 2 2
= 4 43 + 42
= 5
5
44
4+
43
3
= 1
5 2 5 24 +
4
3 2 3 0
= 32
5 16 +
32
3
= 96 240 + 160
15
= 16
15
=
= 2 2
-
EXERCISE 12.4 VOLUME OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 66
= 0
= 0
(0,6)
5 + = 6
3 = 3
1 (6,0)
0 1 3 5
. + = 5 ; = 0 ; = 0 ; = 0
= 0 ; = 5
= 0 ; = 5
= 2 ; = 5
2 = 5 2
= 5 2
=
0
25 10 + 2 5
0
= 25 102
2+
3
3
= 25 5 5 5 2 +1
3 5 3 0
= 125 125 +125
3 0
=
y
5
3
2 y
1 3 5 x
= 0
. + = 6 ; = 3 ; = 0 ;
= (6 )
= 2
= 6
= 36 12 + 2
0
= 36 12 + 2 3
0
= 36 12 2
2 +
3
3
= 36 3 6 3 2 +1
3 3 2 0
= 36 3 6 9 +1
3 27
= [ 36 3 6 9 + 9]
= (9)(12 6 + 1)
= (9)(7)
=
-
EXERCISE 12.4 VOLUME OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 67
. = 4, = 2, = 4; = 4
= 2
= (4 )2
= 4 4
2
= (16 32
2
0
0
+ 16
2 )
= 16 2 322 16
2 0
= 8 4 4 2 1
= .
9. 2 = 4, = ; =
= 2
= ( )2
= ( 2
4)2
2
2
0
= (2 2
2
2
2
+4
162)
= 2 3
6+
5
16 5 2
= 2 2 23
6+
25
16(5)2 2 2
23
16+
25
16(5)2
= 43 1 2
3+
1
5
=
.
-
EXERCISE 12.4 VOLUME OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 68
. = , = 0, = 1; = 1
= 2
= (1 )2
=
2
0
0
(1 )2
= 1 2 + 2
2
0
= [ + 2 +
2
2
4]
= 3
2+ 2 +
2
2
4
= 3
2+ 2
2
4
= 3
4+ 0 4(0) 0 + 2 + 0
= 32
4 2
=
.
-
EXERCISE 12.5 THE WASHER METHOD
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 69
1. = 2 , = 3, = 0;
3. 2 = 4, = ;
= 2 2 2
2
= (2 2
4
22
2
)
= 2 4
162
2
2
= 2 5
802
2
2
= (23 325
802) (23 +
325
802)
= (23 23
5) (23 +
23
5)
=
x y
0 0
a 2a
= 32 2 9
0
= 9 9
0
= 9 2
2
9
0
= 9(9) (9)2
2
9
0
=
X=a
2 = 4
dy x
-
EXERCISE 12.5 THE WASHER METHOD
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 70
5. 2+2 = 2 , =
= 4 2 2 +
0
= 4 2 3
3+
= 4 3 3
3
= 4 23
3
=
7. 2 + 2 = 25 , + = 5 ; = 0
= 25 2 5 2 5
0
=
.
a
o (-a,0) (a,0)
x = b
-
EXERCISE 12.5 THE WASHER METHOD
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 71
9. 2 = 4, 2 = 4;
11. 2 = 8, = 2; = 4
2 = 1
4 =2
4
2
64 = 3
= 4, = 4: (4,4)
= 4 2
2
4
2
4
0
= 4 4
16
4
0
= 22 5
80
4
0
= 2(4)2 +(4)5
80
=
2 = 1
8 = 2 2
8 = 42
= 2, = 4: (2,4)
= 42 43
3
2
0
= 4(2)2 +4(2)3
3
=
RyanRectangle
RyanRectangle
-
EXERCISE 12.6 THE CYLINDRICAL SHELL METHOD
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 72
. 4 = 3 , = 0, = 2, ; = 2
V = 2 2
0
V = 2 2 2
0
3
4 dx
V = 2 [2
2
2
0
4
4]
V = 2 4
4
5
20
20
V = 2 (2)4
4
(2)5
20
20
V = 2 3
5
V =
cubic units
3. = 4 2 , = , = 0
V = 2 3
0
V = 2 4 2 3
0
V = 2 42 3 2 3
0
V = 2 4
32
1
44
1
33
30
V = 2 3 4
4
30
V = 2 (3)3 (3)4
4
30
V =
-
EXERCISE 12.6 THE CYLINDRICAL SHELL METHOD
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 73
5. = , = , =
2
= 2
2
4
=
+ .
7. = 2 , = 0 , = 0
= 2 9 2 9
0
=
.
9. = , = , = 0
= 2
1
= . .
2
Y = 9
(1,0)
(e,1)
X=e
-
EXERCISE 12.6 THE CYLINDRICAL SHELL METHOD
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 74
11. 2 = 8 , = 0 , = 4 ; about = 4
= 2 4 2
8
4
0
=
4 42 3
4
0
=
4
43
3
4
4
0
4
=
13. ( 3 ) 2 + 2 = 9; .
= 8 x 2)3(9 x dx3
0
= 8( ( 9 ( x 3 ) 2
3)
3
2 +27
2
3
3+
9
2( 3)( 9 3 2
3
0
= 8(27 27
2 1)
= 8(27
2)( 1 + )
= 108(
2)
=
RyanRectangle
-
EXERCISE 12.6 THE CYLINDRICAL SHELL METHOD
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 75
15. 2 + 2 = 2 ; = >
= 2 2
= 2 + 2 2 + 2
= 4
: 2 + 2 = 2 = = 2 2
= 4 2 2
= 4
2 2 2
2
2ln + 2 2 +
=
a
a a
a
2 + 2 = 2
-
EXERCISE 12.7 VOLUME OF SOLIDS WITH KNOWN CROSS SECTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 76
. 2 + 2 = 36
=2
2 , = 2
= 22 , = 36 2
=
6
6
= 22
6
6
= 2(3 2)6
6
= .
. 92 + 162 = 144
=1
2
=1
2(2)()
= 2
= 2 28
0
= 2 14492
16
8
0
= .
-
EXERCISE 12.7 VOLUME OF SOLIDS WITH KNOWN CROSS SECTIONS
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 77
.
= (1 )(22)
= 2 (1
2
0
)22
= 2 (1 2
4
2
0
)2
=64
15
= . .
-
EXERCISE 12.8 LENGTH OF AN ARC
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
. = 3
2 = 0 = 5
= 3
2
=3
2
1
2
=
3
2
1
2
= 1 + (
)2
5
0
= 1 + (3
2
1
2)25
0dx
= 1 +9
4
5
0
= .
3. 2
3 + 2
3 + 2
3
= 1 +
1
3
1
3
29
0
=
2
3 + 1
3
2
3
9
0
: 23 =
23 +
23
=
2
3
2
3
9
0
=
1
3
1
3
9
0
= 1
3 3
2
3
2
90
=3
2
= 4 3
2
=
X=0 x=5
78
-
EXERCISE 12.8 LENGTH OF AN ARC
5. = , =
6 =
2
= ; =
1
=
=
=
= 1 +
2 2
6
= 1 + 2
2
6
= .
7. = ,
= (1 )
= ( ) = (1 )
= ( ) = ()
= (1 )
=
= 2 1 2 + 2 sin2 2
0
= 1 2 + sin2 2
0
=
9. = 2 1
= 2 1
= 2
= 2
2 = 4(1 )2
= 4(1 )2 + 422
0
= 2 (1 )2 + 22
0
=
2
6
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy 79
-
EXERCISE 12.9 AREA OF A SURFACE OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1. 2 + 2 = 16 ; = 2 = 4
= 2 4
2
= 16 2
=
1
2 16 2
1
2(2)
=
16 2
= 1 +
2
= 1 +2
16 2
= 16 2 + 2
16 2
=4
16 2
= 2 16 24
16 2
4
2
= 2 44
2
= .
3. 2 = 12 ; = 0 = 3
= 2 3
0
= 12
=
1
2 12
1
2 (12)
=
6
12
= 1 +36
12
= 12 + 36
12
=2 3 + 9
12
= 2 12 2 3 + 9
12
3
0
= 4 3 + 93
0
= . .
5. = 3 ; = 0 = 1
= 32
= 1 + 94
= 2 3 1 + 941
0
= . .
80
-
EXERCISE 12.9 AREA OF A SURFACE OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
7. = 2 ; = 0 =
4
= 2
4
0
= 2(2)
= 1 + 4 2 2
= 2 2 1 + 4 2 2
40
= . .
9. 4 2 = 0 = 2
= 2 2
0
= 2
= 1 + 4^2
= 2 1 + 422
0
= . .
81
-
EXERCISE 12.9 AREA OF A SURFACE OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
13. = ; = 0 ; = 1 ;
= 2 1 + 2 1
0
= 2 1 + 2 = 1
0
= 2 1 + 2( 2/2 ) 10
= 2 1 + 2( )
= +
82
RyanRectangle
-
EXERCISE 13.1 FORCE OF FLUID PRESSURE
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1. =
= (62.5/3)(962)(4)
= 24000
=
=
=
= (62.5
3
)(4)(12
1442)
= (625)(4)
144
= .
3. =
= 1
2 5 3 2 + (
2
3)(3)
=
5. = 50
= 3
=
50 = 1
2 3
1
3
50 =2
2
100 = 2
=
12ft
8ft
5
3
5
3
5
2
3
5
3
5
h
5
3
5
h
5
83
-
EXERCISE 13.1 FORCE OF FLUID PRESSURE
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
7. =
= [()(3)(2)](2)
=
= 6 = major axis
= 4 =
x
0
b
a
y
A=
84
-
EXERCISE 13.2 WORK
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1.
=
= ; = 1
2 , = 40 ; = 0, = 14 10 = 4
40 = 1
2 , = 80
= 804
0
=
3.
=
= ; = 1
10 , = 5 = 0, =
= 50
0
=
85
-
EXERCISE 13.2 WORK
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
5.
=
= 60
= 2 60
= 9(60 )
0
= 9 60 10
0
= 9 60 2 100
= 9 60 2
2
100
= 9 600 50
= .
86
-
EXERCISE 13.2 WORK
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
9.
=
= ; = 10 , = 2, =
2 + 2 = 2 ; = 2 2 ; = 2
= 6 2
2
10 2
= 20 6 2
2
22 2
=
87
-
EXERCISE 13.3 FIRST MOMENT OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1. 2 = 4, = 4
= 1
2 4
4
0
=
= 44
0
= .
3. = 4
=
= 4 4
= 4 2
4
= 2
16
4
4
=
5. 2 = 4 2 = 4
4 = 4
16
64 4 = 0
64 3 = 0
1 = 0, 2 = 4
=1
2 4
4
16
4
=
2
= 4
= 4
88
-
EXERCISE 13.3 FIRST MOMENT OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
7. = 3 [ 9 2 3 3
0
= 3 3 + 3 (3 )23
0
= (3 )3
2(3 + )1
2 (3 )23
0
= 3 9 230
9 2 (3 )23
0
3
0
* = 3 9 23
0
= 92
3 =
3
3 = 9 2 3 = ; =
3
3 =
= 3; =
2
= 0; = 0
= 3 3
20
= 27 2
2
0
= 27 1 + 2
2
2
0
= 27
2+
2
4
2
0= 27
4 =
27
4
* = 9 23
0
= 9 2 @ = 3; = 0
= 2 = 0; = 9
2=
= 1
2
92
3
2
| 30
= 1
2(
2
3)[(9 9)
3
2 (9 0)3
2]
= 1
3 27 = 9
* = (3 )23
0
= 3
=
=(3)3
3 | 3
0
= 0
3
33
3
= 9
= 27
4 9 9
= 27
4 18
= 2772
4
=
[ ]
. = 4 2 , =
= 1
2
2 2
= 1
2 4 2 2 2
3
0
=
89
-
EXERCISE 13.4 CENTROID OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1. + 2 = 6, = 0, = 0
Solving for A
=
= 3
2
6
0
6
0
= [3 2
4] 6
0
= 3 6 36
4
= .
Solving for Solving for
= 6
0 =
6
0
= 3 2
2
6
0 =
1
2 (3
2)
6
0(3
2)
= 3 2
2
6
0 =
1
2 (9 3 +
2
4
6
0)
= [32
2
3
3] 6
0 =
1
2[9
3
22 +
3
12] 6
0
9 = 18 =1
3(3)
= =
Centroid: (2,1)
90
-
EXERCISE 13.4 CENTROID OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
3. = , = 0 = 0
A=
=
0
=
A=2
= ; =
2 = ; =
0
= (
2
0) =
0
=1
2 2
0 =
0
=1
2 2
0 = ; =
=1
2 (
12
2
0) = ; =
=1
2(
2 2
2
2 =
=1
2(
2
2
4) = [ + ]
=
4(2) = + + 0 0
=
2 =
= (
4)(2)
=
8
Centroid:
,
91
-
EXERCISE 13.4 CENTROID OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
7. 2 = 3 , = 2
= (2 3
2
4
0
)
= [2 2
5
5
2]
= [16 64
5]
= 16
5.
= 4
0
= (2 3
2
4
0
)
= (224
0
5
2)
= [2
33
2
7
7
2]
= 5
16[2
3(4)3
2
7(4)
7
2
=5
16[128
3
257
7]
=40
21
= 1
2 2
4
0
= 1
2 [(2)2
4
0
5
2]
= 1
2 (42 3)
4
0
= 1
2 4
33
4
4
4
0
= 10
3
:
,
92
-
EXERCISE 13.4 CENTROID OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
9. 2 + 2 = 25, + = 5
= 25 2 5 5
0
= 25 2 5 + 5
0
5
0
5
0
25 25
0
5 cos = 25 2
5 sin = ; = arcsin
5
5 cos = @ = 5 ; =
2
= 0 ; = 0
= 5 cos 5 cos
2
0
= 25 cos2 cos2 =1 + 2
2
2
0
= 25 1
2 +
1
2 cos 2
2
0
2
0
= 25
4+ 0
=25
4
5 5
0
= 5 = 25
2
2
=25
2
=25
4 25 +
25
2
= 25 25
2
=25 50
4
=25
4( 2)
= 25 2 5 5
0
= 25 2 5 + 25
0
5
0
5
0
= 25 2 = 2
= 1
2
252 32
3
2
52
2+
3
3
= 252
32
3
52
2+
3
3
= 0
3
125
2+
125
3
125
3 0 + 0
= 125
2+
250
3=
375 + 500
6
=125
6
=1
2 25 2
2 5 2
5
0
=1
2 25 2
1
2 5 2
5
0
5
0
=1
2 25
3
3 1/2 25 +
3
3 5
2
= 1
2 125
125
3
1
2 125 +
125
3 125 0
=125
6
:
=
=
125
625 2
4
=125
6
4
25 2
=10
3 2
=
=
125
625 2
4
=10
3 2
,
A B C
25 2
5
x
/2
0
5
0
5
0
5
0 5 0
93
-
EXERCISE 13.5 CENTROID OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1. 2 = ; = 3 ; = 0 ;
= ; =
0 09 3
= 2 9
0
= 2 3 9
0
= = 2 3 +
2
9
0
= 381.70
=
=381.70
152.68
= 2.5
, . ,
. 2 = 4, = 1, = 4, = 0
=
= 2
2
4
1
=2
2 2
4
1
= 4
2
2
4
1
= 16
4
4
1
= 16
4
4
1
= 16
3
4
1
= 16 2
2
1
4
= 8
2
1
4
=15
2
= 2
= 4
2
2
4
1
= 16
4
4
1
= 16
2
4
1
= 16 4
4
1
= 16 3
3
1
4
= 16
33
1
4
=21
4
=
=
15
221
4
= 0,10
7, 0
3
2 =
94
RyanRectangle
RyanRectangle
-
EXERCISE 13.5 CENTROID OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
. 2 = 4 , 2 = 4
= 4 2
2
4
2
4
0
= 4 4
16
4
0
=96
5
= 2 4 +
2
4
2 4
2
4
4
0
=128
3
= =
128
396
5
= ,
,
x y 0 0 1 2 1 4 4
x y
0 0
1/4 1
1 2
4 4
95
-
EXERCISE 13.5 CENTROID OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
. 2 = 4, = = 0
= 2 ( 4
4
0
)
= 26.80829731 .
= 2
= 2 (
4
0
4 +
2) 4
= 64/3
=
= 2.5
y=(0, 2.5, 0)
X Y
0 0
1/4 1
1 2
4 4
X Y 1 1
2 2 3 3
4 4
96
-
EXERCISE 13.6 MOMENT OF INERTIA OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1. 2 + = 6 , = 0 , = 0 ;
0 63 0
= 2
6
0 = 2
6
2
6
0
= 1
2 62 3
6
0
= 1
2 23
4
4
= 1
2 2 6 3
6
4
4
=
3. 3 = , = 8 , = 0 ; =
0
= 2
2
0 = 2(3)
2
0
= 5 2
0
= 6
6 =
26
6
=
5. = 2 , = 0, = 4
0 04 4
= 2
4
0
= 2 4 4
0
= 2 4
2
2
4
0
=
x dy
x dy
(4,4) y
dx
4 - y
97
-
EXERCISE 13.6 MOMENT OF INERTIA OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
7. 2 = 8 , = 2
0 0
1 2 2
0 01 2
2 4 2 4
= 2( )
4
0
= 2(
2
2
8)
4
0
= (3
2
4
8)
4
0
=
9. = 42 , = 4 ;
0 01 4
0 01 4
= 2( )
= 2(4 42)
1
0
Iy=1
5
X1 X2
dy y
(0,0)
(1,4)
= 42
= 4
dx
98
-
EXERCISE 13.6 MOMENT OF INERTIA OF A PLANE AREA
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
11. 2 = 8 , = 0 , = 4 , with respect to = 4
= 4 2
2
8
4
0
=1
8 162 83 + 4
4
0
=1
8
163
3 24 +
5
5
0
4
=
13. = , = 2 , + = 6, = 0
0 01 1
0 01 2
0 01 5
2 2 2 4 2 4
=
=
=
+ = 6
= 2
(6 2) 6 3
=
99
-
EXERCISE 13.7 MOMENT OF INERTIA OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
1. = 2 , = 0 , = 4 ;about = 0
= 2 3 2 0
4
0
= 4 7
2 4
0
= 4 2
92
9
0
4
= 4 2 4
92
9
2 0 9
2
9
0
4
= 4 1024
9
=
3. + = , = 0 , = 0 ;about the y-
axis
= 2 3
0
0
=2
3 4
0
=2
3 4
0
0
=2
4
4
0
5
5
09
=2
5
4
5
5
=2
5
20
=
100
-
EXERCISE 13.7 MOMENT OF INERTIA OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
5. 2 + 3 = 6 , = 0 , = 0 ; about the x-
axis
=
2
6 2
3
4
0 3
0
=
2
1641923+86421728+1296
81
3
0
=
7. 2 = 3 , = ; about = 0
= 2 3 3
3
0
= 2 7
2 3 4 3
0
= 2 3 7
2 43
0
3
0
= 2 54 243
5
=
9. = 4 , = , = 1 ; about = 0
= 2 3
4
2
1
= 2 42 4 2
1
= 2 4 3
3
1
2
5
5
1
2
= 2 28
3
31
5
=
101
-
EXERCISE 13.7 MOMENT OF INERTIA OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
11. = 2 , = 2 ;about the y-axis
= 2 3 2 2
2
0
= 2 24 5 2
0
= 2 2 4 52
0
2
0
= 2 2 5
5
0
2
6
6
0
2
= 2 64
5
32
3
=
13. = 3 , = 1 , = 0 ; about = 1
= 2 + 1 3 3 0
1
0
= 2 6 + 35 + 34 + 3 1
0
= 2 7
7+
6
2+
35
5+
4
4
0
1
=
102
-
EXERCISE 13.7 MOMENT OF INERTIA OF A SOLID OF REVOLUTION
DIFFERENTIAL & INTEGRAL CALCULUS | Feliciano & Uy
15. = 2 , = 1 , = 0 ; about = 2
= 2 2 3 2
1
0
= 4 8 122 + 63 4 1
0
= 4 42 43 +34
2
5
5
0
1
=
103
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