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Chapter 13: Q 49 (page 1004)

Evaluate each of the double integrals in Exercises 37-54as iterated integrals.

localid="1650381493840" ∫∫Ry2sinxdA,

wherelocalid="1650381496900" R=x,y|0≤x≤πand0≤y≤3

Short Answer

Expert verified

The value of double integral is :-

∫∫Ry2sinxdA=18

whereR=x,y|0≤x≤πand0≤y≤3

Step by step solution

01

Step 1. Given Information 

We have given the following double integral :-

∫∫Ry2sinxdA,

where R=x,y|0≤x≤πand0≤y≤3

We have to evaluate this double integral.

02

Step 2. Use iterated integrals

The given double integral is :-

∫∫Ry2sinxdA,

where R=x,y|0≤x≤πand0≤y≤3

Then by using Fubini's Theorem, we can writ this double integral as following :-

∫∫Ry2sinxdA=∫03∫0πy2sinxdxdy

Then by using iterated integrals, we have :-

∫03∫0πy2sinxdxdy=∫03∫0πy2sinxdxdy

Now we can solve this integral as following :-

∫03∫0Ï€y2sinxdxdy=∫03-y2cosxÏ€0dy=∫03-y2³¦´Ç²õÏ€--y2cos0dy=∫03-y2-1+y21dy=∫03y2+y2dy=∫032y2dy=2y3330=2333-0=2×9=18

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