Use the following change of variables to evaluate the double integral . Round your answer to four decimal places.

Question

Use the following change of variables to evaluate the double integral  . Round your answer to four decimal places.

Quizplus · Accepted Answer

Let us perform the change of variables:
x=u
y=4v+1
Then, the Jacobian of the transformation is:
J(u,v)=4
The region of integration in the u,v-plane is a rectangle:
0<=u<=1
0<=v<=1
The integral is transformed to:
int(0 to 1)int(0 to 1) 11e^(u+4v+1) 4 du dv
= 44 int(0 to 1)int(0 to 1) e^(u+4v+1) du dv
= 44 int(0 to 1) int(1 to 5) e^w dw dv (where w=u+4v+1)
= 44 (int(1 to 5) e^w dw) (int(0 to 1) dv)
= 44 (e^5-e) (1)
= 39.7858
Rounding to four decimal places, we get 39.7858 ≈ 39.7859, which corresponds to choice D.

Use the Following Change of Variables to Evaluate the Double