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6.Definite Integral
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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
−=∫
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&����
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………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
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
Calculus 1
5
LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL
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
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$ก����� x = a cos t + a t sin t ���
y = a sin t - a t cos t %�ก t = 0 �+� t = 2π
Calculus 1
8การหาพืน้ที่การหาพืน้ที่การหาพืน้ที่การหาพืน้ที่
ก����!�,�����������A�G1�� ���ก�����)Xก��(��ก������ !�"�%&�ก�'�(� *+��������� 2 ��ก[\� 1�� 1. ก������������ ��!�����"#���$�������%!&%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(���(������+�(���(�� '������)������#��,
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
Calculus 1
10����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� y = 5x - x5 ��� y = 4x
����� [-1 , 1] ����������� %���!�,���(����\��� ��\����]'���$'��)�A�G1�� y2 = 2x ����A����
x - y = 4
Calculus 1
11
NOTE LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL
Calculus 1
12