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In Exercises 59–62, evaluate the double integral over the specified region.

Ωxex3dA

Short Answer

Expert verified

Value of the integral Ωxex3dAover the rectangular region is,

02y2xex3dxdy=e8-13.

Step by step solution

01

Step 1. Given information.

We have given integral,

Ωxex3dA

over the triangular region with vertices (0, 0), (2, 0) and (2, 2).

02

Step 2. Explanation.

Drawing the rectangular region,

From the above region,

dA=dxdy

and yx2,0y2

Hence,

02y2xexdxdy

Changing the order of integration,

02y2xex3dxdy=020xxex3dydx

Evaluating the integral,

020xxex3dydx=02x2ex3dx

Substituting x3=udu=3x2dxx2dx=13du

02x2ex3dx=08eu3du

=13eu08

=e8-13

03

Step 3. Conclusion.

Hence, value of the integral Ωxex3dAover the rectangular region is,

02y2xex3dxdy=e8-13.

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