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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Think About It Explain why you need to change the order of integration to evaluate the double integral. Then evaluate the double integral.

(a) 0 3 y 3 e x 2 d x d y

(b) 0 2 x 2 e y 2 d y d x

(a)

To determine

The reason for which it is required to change the order of integration to solve the double integral 03y3ex2dxdy and then find the value of this integral.

Explanation

Given Information:

The provided double integral is 03y3ex2dxdy.

Consider the double integral 03y3ex2dxdy.

From the limits of integration, the inner limit of integration is yx3, which implies that the region R is bounded on the left by the line x=y and on the right by the line x=3.

Also since 0y3, the region lies above the x-axis as shown in figure below,

For the expression ex2 form the provided order of integration no antiderivative is found.

So, in order to solve the double integral 03y3ex2dxdy, change the order of integration such that x is the outer variable.

So, the constant bound of integration for x would be given by,

0x3

Which is the outer limit of integration.

The constant bound of integration for y would be given by,

0yx

Which is the inner limit of integration.

Thus the graph for the change order of integration would be given by,

From the above graph the with x as the outer variable the integral can be written as,

030xex2dydx

Now determine the value of the above integral

(b)

To determine

The reason for which it is required to change the order of integration to solve the double integral 02x2ey2dydx and then find the value of this integral.

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