12
Calculus 1 1 การหาปริพันธจากัดเขต ทฤษฎีพื้นฐานสาหรับอินทิกรัลแคลคลัส f ] b , a [ ) x ( F ) x ( f กก f (x) b a dx ) x ( f ) a ( F ) b ( F ) x ( F dx ) x ( f b a b a = = คุณสมบัติของปริพันธจ ากัดเข คุณสมบัติของปริพันธจ ากัดเข คุณสมบัติของปริพันธจ ากัดเข คุณสมบัติของปริพันธจ ากัดเขต a, b, c, f (x) , g (x) [a , b] 1. ) x ( f b a dx = 0 2. ) x ( f b a dx = - ) x ( f a b dx 3. ) x ( kf b a dx = k ) x ( f b a dx k 4. )] x ( g ) x ( f [ b a ± dx = ) x ( f b a dx ± ) x ( g b a dx 5. ) x ( f b a dx = ) x ( f c a dx + ) x ( f b c dx a < c < b บททีบททีบททีบทที่ 6 ปริพันธ ปริพันธ ปริพันธ ปริพันธจ ากัดเขตและการประยุกต ากัดเขตและการประยุกต ากัดเขตและการประยุกต ากัดเขตและการประยุกต (Definite Integral and Applications) (Definite Integral and Applications) (Definite Integral and Applications) (Definite Integral and Applications) จาไวนะครับสตรการหาปริพันธจากัดเขต ) a ( F ) b ( F dx ) x ( f b a =

Definite Integral

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Page 1: Definite Integral

Calculus 1

1

การหาปริพันธจํากัดเขต ทฤษฎีพื้นฐานสําหรับอินทิกรัลแคลคูลัส

��� f �����ก������������������� ]b,a[ ������ )x(F ���ก������ !�"�

#$�%&�ก�'�(�(�� )x(f ���� ก���������� ��ก������� f (x) %���'��) ∫b

a

dx)x(f

*+��%���#'�'���,

)a(F)b(F)x(Fdx)x(fb

a

b

a

−==∫

คณุสมบตัขิองปริพนัธจาํกัดเขคณุสมบตัขิองปริพนัธจาํกัดเขคณุสมบตัขิองปริพนัธจาํกัดเขคณุสมบตัขิองปริพนัธจาํกัดเขตตตต

��� a, b, c, ���1��1���� ��� f (x) , g (x) �����ก���������������� [a , b]

1. ∫ )x(fba dx = 0

2. ∫ )x(fba dx = - ∫ )x(fa

b dx

3. ∫ )x(kfba dx = k ∫ )x(fb

a dx A&�����1��1���� k �' B

4. ∫ )]x(g)x(f[ba ± dx = ∫ )x(fb

a dx ± ∫ )x(gba dx

5. ∫ )x(fba dx = ∫ )x(fc

a dx + ∫ )x(fbc dx �$��� a < c < b

บทที ่บทที ่บทที ่บทที ่6666 ปรพินัธปรพินัธปรพินัธปรพินัธจาํกดัเขตและการประยกุตจาํกดัเขตและการประยกุตจาํกดัเขตและการประยกุตจาํกดัเขตและการประยกุต

(Definite Integral and Applications)(Definite Integral and Applications)(Definite Integral and Applications)(Definite Integral and Applications)

จําไวนะครับสูตรการหาปริพันธจํากัดเขต

)a(F)b(Fdx)x(fb

a

−=∫

Page 2: Definite Integral

Calculus 1

2

�������� %���1�� ∫ −+2

0

3 dx)3x2x(

��� � ∫∫∫∫ −+=−+2

0

2

0

2

0

32

0

3 dx3xdx2dxxdx)3x2x(

[ ]

[ ] [ ]

[ ] [ ]

2

644

230404

16

023)0()2(4

)0(

4

)2(

x32

x2

4

x

2244

20

2

0

22

0

4

=

−+=

−−+

−=

−−−+

−=

+

=

�������� %���1�� ( )∫− −

2

1 3

2

dx2x

x

��� � ��� 2xu 3 −= %�#'� )2x(ddu 3 −=

dx

x3

du

dxx3du

2

2

=

=

��1���G%�)� %�#'�

( )

[ ]

[ ]

[ ]

[ ])3ln()6ln(3

1

)2)1ln(()2)2ln((3

1

)2xln(3

1

uln3

1

duu

1

3

1

x3

du

u

xdx

2x

x

33

2

13

21

2

1

2

1 2

22

1 3

2

−−=

−−−−=

−=

=

=

=−

−−

∫∫

���กA&�1�H�ก������ !�"�%&�ก�'�(�1�������

∫ dx)x(f ��$� "����ก��������)�����, ���#'�1&��������ก��

F(x) ����#$�������ก c �������� F(b) G')�� x = b � F(x) �����F(a) G')�� x = a � F(x) ������1�� F(b) J F(a) %�#'����1&����

Page 3: Definite Integral

Calculus 1

3

�������� %���1�� ∫π

0

2 dx)xcos(x5

��� � ��� 2xu =

%�#'� )x(ddu 2=

dx

x2

du

xdx2du

=

=

��1���G%�)� %�#'�

[ ]

[ ]

[ ]

[ ]

[ ]

0

002

5

0sinsin2

5

)0(sin)(sin2

5

xsin2

5

usin2

5

du)ucos(2

5x2

du)ucos(x5

dx)ucos(x5dx)xcos(x5

22

02

0

0

0

00

2

=

−=

−π=

−π=

=

=

=

=

=

π

π

π

π

ππ

∫∫

NOTE………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

Page 4: Definite Integral

Calculus 1

4�������� %���1���� !�"�%&�ก�'�(�(����ก������#��,

1. ∫ ++1

0

4232 dx)x2x)(x4x3( 2. ∫π

4

0

3 dxx2cosx2sin

3. ∫−3

1 4dx

x

5x3 4. ∫ +++1

0 24

3

dx1xx

xx2

5. ∫2

0

x2 dx7x3 6. ∫ ++

3

1

x2x dxe)1x(2

LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

Page 5: Definite Integral

Calculus 1

5

LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

Page 6: Definite Integral

Calculus 1

6

แบบฝกหัดแถมครับ ต้ังใจทํานะครับจะไดไมติด F

1. ∫ −−+−

3

2

32 dx)xx4x35(

2. ∫ −−−−

0

5

732 dx)4x20x5()4x3(

3. ∫−

π

π

2

4

dxx2sin6

4. ∫−

15

10

2x3 dx5

5. ∫ −−

1

1

x dx)e2x4

(

NOTE LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

Page 7: Definite Integral

Calculus 1

7

การหาความยาวสวนโคงการหาความยาวสวนโคงการหาความยาวสวนโคงการหาความยาวสวนโคง

����ก�������������������

�����'��$���ก x

AR����� 1 L = ∫ 2ba )

dxdy

(1+ dx

�����'��$���ก y

AR����� 2 L = ∫ 2dc )

dydx

(1+ dx

����ก��������� ������������ก����ก�ก��

x = f (t) , y = g (t) a ≤ t ≤ b 1��

AR����� 3 L = ∫ 22ba )

dtdy

()dtdx

( + dt

����������� %���1��$)��(��A��G1�� x = 2/3)1y(32

− %�ก y = 1 �+� y = 4

����������� %���1��$)��(��A��G1�� 1x2yx8 62 =− %�ก%X' (1 , 83

) �+�%X'

(2 , 32

129)

����������� %���1��$)��(��A��G1�� )ee(21

y xx −+= %�ก x = -1 �+� x = 1

����������� %���1��$)��(��A��G1��*+��$�A$ก��� ��������A� $��� x = 2t ��� 2/3t

32

y = �$��� 5 ≤ t ≤ 12

����������� %���1��$)��(��A��G1��*+��$�A$ก����� x = a cos t + a t sin t ���

y = a sin t - a t cos t %�ก t = 0 �+� t = 2π

Page 8: Definite Integral

Calculus 1

8การหาพืน้ที่การหาพืน้ที่การหาพืน้ที่การหาพืน้ที่

ก����!�,�����������A�G1�� ���ก�����)Xก��(��ก������ !�"�%&�ก�'�(� *+��������� 2 ��ก[\� 1�� 1. ก������������ ��!�����"#���$�������%!&%ก� x ����%ก� y

1.1 ก����!�,���*+�����$���'��)�A�G1������ก x ��� f �����ก�����������������]' [a , b] ��� A ���!�,���(����\��� ��\*+���Rก���$���'��)�A�G1�� y = f(x) �ก x �A���� x = a ����A���� x = b

��� f (x) ≥ 0 A&������Xก x ε [a , b] %�#'�

A = ∫ ba f (x) dx (!�,����)R������ก x)

��� f (x) ≤ 0 A&������Xก x ε [a , b] %�#'�

A = ∫ ba -f (x) dx (!�,����)R�����ก x)

1.2 ก����!�,���*+�����$���'��)�A�G1������ก y ��� g �����ก�����������������]' [c , d] ��� A ���!�,���(����\��� ��\*+���Rก���$���'��)�A�G1�� x = g (y) �ก y �A���� y = c ����A���� y = d

��� g (y) ≥ 0 A&������Xก y ε [c , d] %�#'�

A = ∫ ba g (y) dy (!�,����)R����(���ก y)

��� g (y) ≤ 0 A&������Xก y ε [c , d] %�#'�

A = ∫ ba -g (y) dy (!�,����)R����*��)�ก y)

� "�ก����!�,���*+�����$���'��)�A�G1������ก x ���� �ก y $�(�,��'���, 1. �(�)ก��%�กA$ก�����G%�)�ก&��'!�A���(� 2. 'R1&�A��� �����!�,������$���'��)�A�G1��ก���ก x �����ก y 3. ��'����A�������)$_�_����������`�)������!�,������G%�)�ก&��'�����,�a�กก��

�ก x �����ก y 4. ��(���(� [a , b] ���� [c , d] %�กG%�)� 5. ����ก���AR�� 6. 1&��\!�,����������ก��%�ก(���(������+�(���(�� '������)������#��,

Page 9: Definite Integral

Calculus 1

9

����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� y = x2 - 4 ��� �ก x ����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� y = x3 - 6x2 + 8x ��� �ก x ����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� x = 4 - y2 �ก y

�A���� y = -1 ��� y = 2 2. ก�������������������&�����$�������

2.1 ก����!�,�������)R���������A�G1�� ���$��ก��)�!�"�ก���ก x ��� f ��� g �����ก������$�1��$��������������]' [a , b] G')���

f (x) ≤ g (x) �Xก x ε [a , b] A ���!�,���(����\��� ��\*+���Rก���$���'��)�A�G1�� y1 = f (x) �A�G1�� y2 = g (x) �A���� x = a ��� x = b ����

A = ∫ ba )]x(f)x(g[ − dx �$��� g (x) ≥ f (x)

2.2 ก����!�,�������)R���������A�G1�� ���$�1��$�ก��)�!�"�ก���ก y ��� f ��� g �����ก������$�1��$��������������]' [c , d] G')���

f (y) ≤ g (y) �Xก y ε [c , d] A ���!�,���(����\��� ��\*+���Rก���$���'��)�A�G1�� x1 = f (y) ����A�G1�� x2 = g (y) �A���� y = c ����A�G1�� y = d ����

A = ∫ ba )]y(f)y(g[ − dy �$��� g (y) ≥ f (y)

��ก����������������&�����$������� ��*������#�����+,���

1. �(�)ก��%�กA$ก�����G%�)�ก&��'!�A���(� 2. 'R1&�A�����������!�,������$���'��)�A�G1���'ก���A�G1���' �����%X'��'(���A�G1��

��,�A���!���������(���(� 3. ��'����A�������)$_�_����������`�)�����!�,������G%�)�ก&��' 4. ����ก���AR�� 5. 1&��\!�,����������ก�� %�ก(���(������+�(���(�� '������)������#��,

����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� y = 2x2 ��� y = 3x

Page 10: Definite Integral

Calculus 1

10����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� y = 5x - x5 ��� y = 4x

����� [-1 , 1] ����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� y2 = 2x ����A����

x - y = 4

Page 11: Definite Integral

Calculus 1

11

NOTE LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

Page 12: Definite Integral

Calculus 1

12