[WWW.toanPhoThong.tk] Bai Tap Ung Dung Cua Tich Phan

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WWW.ToanPhoThong.TKCHUYN

NG DNG CA TCH PHNA. TCH PHN CHA GI TR TUYT I Phng php gii ton 1. Dng 1b

I =

a Gi s cn tnh tch phn , ta thc hin cc bc sau Bc 1. Lp bng xt du (BXD) ca hm s f(x) trn on [a; b], gi s f(x) c BXD:

f(x) dxx1 0

x f(x)b

a

+x1

-

x2 0x2

+

b

b

I = Bc 2. Tnh

f(x) dx = f(x)dx - f(x)dx + f(x)dxa a x1 x2 2

.

I = V d 1. Tnh tch phn Bng xt du1

x- 3

2

- 3x + 2 dx . Gii - 3 +2

x x - 3x + 22

1 0

-

2 0 59 2 .

I =

( x- 3

2

- 3x + 2) dx p 2

( x1

2

- 3x + 2) dx =

I = V d 2. Tnh tch phnp 2

0

5 - 4cos2 x - 4sin xdx . Giip 2

I = Bng xt du

0

4sin2 x - 4sin x + 1 dx = x 2sin x - 1 0 p 6 0

2sin x 0

1 dx .

+

p 2

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WWW.ToanPhoThong.TKp 6 p 2

I =-

( 2sin x 0

1) dx + ( 2sin x - 1) dx = 2 3 - 2 p 6

p 6 .

2. Dng 2b

I = Gi s cn tnh tch phn Cch 1.b

[ f(x)a b

g(x) ] dx , ta thc hin:b

I = Tch

[ f(x)a

g(x) ] dx =

f(x) dx g(x) dxa a

ri s dng dng 1 trn.

Cch 2. Bc 1. Lp bng xt du chung ca hm s f(x) v g(x) trn on [a; b]. Bc 2. Da vo bng xt du ta b gi tr tuyt i ca f(x) v g(x).2

I = V d 3. Tnh tch phn Cch 1.2

(- 1

x - x - 1 ) dx . Gii2 2

I = =-

(- 1 0 - 1

x - x - 1 ) dx =2 1

x dx - x - 1 - 1 2

1 dx 1 )dx

xdx + xdx + (x 2 0 - 1

1 )dx 1 2

x =2 Cch 2. Bng xt du

x + 2

0 2 2

- 1

(x 1 2

0

x + - x 2 - 12

x - x =0 2 1 . 2 + +2 1

x x x10

1

0 01

1 + 0

I =

( - x + x - 1

1) dx + ( x + x - 1) dx + ( x - x + 1) dx0 2 1 0 2 +x1 = 0.

= - x -0 1 + ( x - x )

Vy I = 0 .

3. Dng 3

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2

WWW.ToanPhoThong.TKb b

I = tnh cc tch phn thc hin cc bc sau:

max { f(x),a

g(x) } dx v

J =

min { f(x),a

g(x) } dx , ta

Bc 1. Lp bng xt du hm s h(x) = f(x) - g(x) trn on [a; b]. Bc 2. + Nu h(x) > 0 th max { f(x), g(x) } = f(x) v min { f(x), g(x) } = g(x) . + Nu h(x) < 0 th max { f(x), g(x) } = g(x) v min { f(x), g(x) } = f(x) .4

I = V d 4. Tnh tch phn

max { x0 2

2

+1 , 4x - 2} dx .

Gii 2 ( ) t h(x) = ( x + 1) - 4x - 2 = x - 4x + 3 . Bng xt du x h(x)1

0 +2

1 03 1

3 0

4 +4 3

I =

( x0

+ 1) dx + ( 4x - 2) dx + ( x2 + 1) dx =2

80 3

.

I = V d 5. Tnh tch phn

min { 3 ,x 0

4 - x } dx .

Gii x ( 4 - x ) = 3x + x - 4 h(x) = 3 t . Bng xt du x h(x)1 2

0

1 0x

2 +1 2

I =

3x dx + ( 4 - x ) dx =0 1

3 x2 2 5 + 4x = + ln 3 0 2 ln 3 2 1

.

B. NG DNG CA TCH PHNI. DIN TCH HNH PHNG

1. Din tch hnh thang cong

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WWW.ToanPhoThong.TKCho hm s f(x) lin tc trn on [a; b]. Din tch hnh thang cong gii hn bi ccb

ng y = f(x), x = a, x = b v trc honh l:

S = f(x) dxa

.

Phng php gii tonBc 1. Lp bng xt du hm s f(x) trn on [a; b].b

Bc 2. Da vo bng xt du tnh tch phn

f(x) dxa

.

, x = e v Ox. V d 1. Tnh din tch hnh phng gii hn bi y = ln x, x = 1 Gii [1 ] ln x 0 " x ; e Do nn:e e

S=

ln x dx = ln xdx = x ( ln x 1 1

1)

e 1

Vy S = 1 (vdt).

=1 .

2 , x=0 , x = 3 v V d 2. Tnh din tch hnh phng gii hn bi y = - x + 4x - 3 Ox. Gii Bng xt du x 0 1 3 y 0 + 0 1 3 2

S=-

( - x0 3

+ 4x - 3) dx + ( - x2 + 4x - 3) dx1 1 3

x x3 8 2 =- + 2x + 3x + + 2x2 + 3x = 3 0 3 1 3. 8 S= 3 (vdt). Vy

2. Din tch hnh phng 2.1. Trng hp 1Cho hai hm s f(x) v g(x) lin tc trn on [a; b]. Din tch hnh phng gii hn bib

cc ng y = f(x), y = g(x), x = a, x = b l:

S = f(x) - g(x) dxa

.

Phng php gii ton WWW.ToanPhoThong.TK 4

WWW.ToanPhoThong.TKBc 1. Lp bng xt du hm s f(x) - g(x) trn on [a; b].b

Bc 2. Da vo bng xt du tnh tch phn

f(x) a

g(x) dx .

2.2. Trng hp 2Cho hai hm s f(x) v g(x) lin tc trn on [a; b]. Din tch hnh phng gii hn bib

S = f(x) - g(x) dx y = f(x), y = g(x) a cc ng l: . a , b Trong l nghim nh nht v ln nht ca phng trnh f(x) = g(x)( a a < b b) .

Phng php gii tonBc 1. Gii phng trnh f(x) = g(x) . Bc 2. Lp bng xt du hm s f(x) - g(x) trn on [ a; b] .b

Bc 3. Da vo bng xt du tnh tch phn

f(x) a

g(x) dx .

V d 3. Tnh din tch hnh phng gii hn bi cc ng: y = x3 + 11x - 6 , y = 6x2 , x = 0 , x = 2. Gii 3 2 3 h(x) = (x + 11x 6) 6x = x 6x2 + 11x - 6 t h(x) = 0 x = 1 x = 2 x = 3 (loi). Bng xt du x 0 h(x)1

2

1 0

+

2 0

S=-

( x0 4

3

- 6x + 11x - 6) dx + ( x3 - 6x2 + 11x - 6) dx2 1 2 1 2

x 11x x4 11x2 5 3 3 =- 2x + 6x + 2x + - 6x = 4 0 4 1 2 2 2. 5 S= 2 (vdt). Vy3 , y = 6x2 . V d 4. Tnh din tch hnh phng gii hn bi cc ng y = x + 11x - 6

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WWW.ToanPhoThong.TKGii 2 t h(x) = (x + 11x - 6) - 6x = x - 6x + 11x - 6 h(x) = 0 x = 1 x = 2 x = 3 .3 2 3

Bng xt du x 1 h(x) 02 3 3

+

2 03

3 0

S=

( x1 4

- 6x + 11x - 6) dx 2 2 2

( x2

- 6x2 + 11x - 6) dx3

x 11x = - 2x3 + - 6x 4 2 1 Vy S= 1 2 (vdt).

x4 11x2 1 - 2x3 + - 6x = 4 2 2 2.

Ch :1) Nu hnh phng c gii hn t 3 ng tr ln th phi v hnh, tuy nhin hu ht rt kh xc nh ng min phng cn tnh din tch (c th v th m thi i hc khng ra). 2) Nu trong khong dng cng thc:

( a; b)b

phng trnh f(x) = g(x) khng c nghim th ta c thb

f(x) a

g(x) dx =

f(x) a

g(x) dx

3) Nu tch din tch hnh phng gii hn bi x = f(y) v x = g(y) th ta gii nh trn nhng nh i vai tr x cho y (xem v d 9).3 V d 5. Tnh din tch hnh phng gii hn bi y = x , y = 4x .

Gii Phng trnh honh giao im: x3 = 4x x = - 2 x = 0 x = 20 2 3

S=

( x- 2

- 4x ) dx +0

( x0

3

- 4x ) dx

x4 x4 2 = 2x + - 2x2 =8 4 - 2 4 0 Vy S = 8 (vdt).

2

.

2 V d 6. Tnh din tch hnh phng gii hn bi y = x - 4 x + 3 v trc honh. Gii Phng trnh honh giao im:

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WWW.ToanPhoThong.TKx2 - 4 x + 3 = 0 t2 t =1 x = 1 x = 3 t=3 3

- 4t + 3 = 0 , t= x 0 x = 1 x = 3 3 0

S=

x- 3

2

- 4 x + 3 dx = 2 x2 - 4x + 3 dx

3 1 2 2 = 2 x 4x + 3 dx + x 4x + 3 dx ( ) ( ) 1 0 1 3 3 3 x x 16 = = 2 - 2x2 + 3x + - 2x2 + 3x 0 3 1 3 3

16 S= 3 (vdt). Vy

.

2 V d 7. Tnh din tch hnh phng gii hn bi y = x - 4x + 3 v y = x + 3 . Gii Phng trnh honh giao im: x + 3 0 x=0 x2 - 4x + 3 = x + 3 x=5 2 2 x - 4x + 3 = - x - 3 x - 4x + 3 = x + 3 . Bng xt du x 0 1 3 5 + 0 0 + x2 - 4x + 3 1 3 2 2 5

S=

( x0 3

- 5x ) dx + ( - x + 3x - 6) dx + ( x2 - 5x ) dx1 3 2 1 3 2 3 3 5

x 5x -x 3x x 5x2 109 = + + 6x + = 3 3 3 2 2 2 6 0 1 3 109 S= 6 (vdt). Vy

.

2 V d 8. Tnh din tch hnh phng gii hn bi y = x - 1 , y = x + 5 . Gii Phng trnh honh giao im: x2 - 1 = x + 5 t2 - 1 = t + 5 , t= x 0 t= x 0 t= x 0 t2 - 1 = t + 5 x = 3 t=3 2 t - 1= - t - 5 3 3

S=

- 3

x2 - 1 -

(

x + 5) dx = 2 x2 - 1 0

(

x + 5) dx

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WWW.ToanPhoThong.TKBng xt du x x - 12 1 3 2

0 1

1 0

3 +

S=2

( - x0 3

- x - 4) dx + ( x2 - x - 6) dx2 1 3

-x x x3 x2 73 =2 4x + 6x = 3 3 2 2 3 0 1 73 S= 3 (vdt). Vy

.

, y= V d 9. Tnh din tch hnh phng gii hn bi y = x, y = 0 Gii 2 2 Ta c: y = 2 - x x = 2 - y , x 0. Phng trnh tung giao im: y =1

2 - x2 .

2 - y2 y = 1. 2 - y2 - y ) dy

S=

2 - y - y dy =1

2

1 = 2cos tdt - ydy = t + sin2t 2 0 02

0 p 4

(0

1

(

)

p 4 0

y2 2

1

0

p S= 4 (vdt). Vy Cch khc:

.

1 V hnh ta thy S bng 8 din tch hnh trn bn knh R =

2 nn

S=

1 2 p pR = 8 4.

II. TH TCH KHI TRN XOAY 1. Trng hp 1Th tch khi trn xoay do hnh phng gii hn bi cc ng y = f(x) 0" x [ a;b ] ,b

y = 0 , x = a v x = b (a < b) quay quanh trc Ox l: a . 2 2 2 V d 1. Tnh th tch hnh cu do hnh trn (C) : x + y = R quay quanh Ox. Gii 2 x = R 2 x = R . Honh giao im ca (C) v Ox l 2 2 2 2 2 2 Phng trnh (C) : x + y = R y = R - x x3 4pR 3 V = p ( R - x ) dx = 2p ( R 2 - x2 ) dx = 2p R 2x = 3 3 . - R 0 02 2 R R R

V = p f 2(x)dx

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WWW.ToanPhoThong.TKV = 4pR 3 3 (vtt).

Vy

2. Trng hp 2

Th tch khi trn xoay do hnh phng gii hn bi cc ng x = g(y) 0" y [ c;d ] ,d c x = 0 , y = c v y = d (c < d) quay quanh trc Oy l: . 2 2 x y (E) : 2 + 2 = 1 a b V d 2. Tnh th tch hnh khi do ellipse quay quanh Oy. Gii y2 = 1 y = b 2 Tung giao im ca (E) v Oy l b . 2 2 2 2 x y ay (E) : 2 + 2 = 1 x2 = a2 a b b2 Phng trnh R 2 a2y2 2 a2y2 2 a2y3 4pa2b V = p a dy = 2 p a dy = 2 p a y = 2 2 b b 3 . 3b2 - b 0 0 b b

V = p g2(y)dy

Vy

V =

4pa2b 3 (vtt).

3. Trng hp 3

Th tch khi trn xoay do hnh phng gii hn bi cc