16.09b Change of Variables Continued y Ex. 3:
e
x y x y
x
R
dA , where R is the trapezoidal region at right:
R
The original function is difficult to integrate, so we will try a change of variables with u x y and v x y . These equations define a change of variables T 1 from the xy plane to the uv plane. We need to go the other way, so we solve for x and y . 1 u v y 2 1 1 1 1 y u vv u v 2 2 2 2 1 y u v 2
u x y v x y v x y u v 2x 1 x u v 2 x u The Jacobian is: y u
v
x 1 v 2 y 1 v 2
1 2 11 1 1 4 4 2 2 R1 : y 0 u x, v x We can determine the region S as follows: v u y R2 : y x 2 R1
y x 1
R4
y x2
R
T v2
R3 : x 0 u y , v y v u
T 1
v
v u
v 2
R2
R3
S3
2 x y
x
S2
S S4
S1
R4 : y x 1
vu
1 x y
v 1
u
Hanford High School Calculus III
v 1
16.09b Change of Variables Continued – G. Kelly
1 x 2
1 u 2 0 x2 1 0 u 2 2 2 0u2 4 2 u 2 2 y 1 2 v 1 2 v 1 1 v 2 0 x 1 1 0 u 1 1 2 0 u 1 2 1 u 1 Page 1 of 2
The integral then becomes: x y
u
x y e dA e v R
S
2
1
v
v
x, y du dv u, v
Absolute value of the Jacobian
1 e du dv 2 u v
u v
1 2 u ve v dv 2 1 u v
1 2 e e1 v dv 2 1 2
1 1 e e1 v 2 2 2 1
3 e e1 4
Change of Variables for Triple Integrals is discussed on page 1055.
x u x, y, z y u, v, w u z u
The formula for the Jacobian is:
x v y v z v
x w y w z w
The formula for triple integrals is:
x, y , z
f x, y, z dV f x u, v, w , y u, v, w , z u, v, w u, v, w R
du dv dw
S
Neither of these formulas will be on the formula quizzes.
Hanford High School Calculus III
16.09b Change of Variables Continued – G. Kelly
Page 2 of 2