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Chapter 13: Q. 13 (page 1082)

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.

鈭�09鈭�y311+x3dxdy

Short Answer

Expert verified

鈭�0x2鈭�0311+x3dxdy=13ln28

Step by step solution

01

Draw the region

The region determined by the limits of the given iterated integral is shown below,

02

Reversing the order of integration  

From the above diagram, reversing the order of integration.

鈭�09鈭�y311+x3dxdy鈬�鈭�0x2鈭�0311+x3dxdy

03

Evaluate the integral

I=鈭�0x2鈭�0311+x3dxdyI=鈭�0x2dy鈭�0311+x3dxI=鈭�03x21+x3dx

Substitute,

1+x3=t3x2dx=dtdx=13x2dtWhenx=0,t=1+0=1Whenx=3,t=1+33=28

Therefore,

I=13鈭�1281tdtI=13lnt128I=13ln28

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