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  vv  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 11 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  x2

R

T v2

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

vu

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 x2 1 0  u  2  2 2 0u2 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  e1  v dv 2 1 2

1 1   e  e1   v 2 2 2 1 

3  e  e1  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

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