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Chapter 13: Q. 35 (page 1083)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.

∫0π∫02π∫05p2sinϕdpdθdϕ

Short Answer

Expert verified

∫0π∫02π∫05p2sinϕdpdθdϕ=5003π

Step by step solution

01

Given

We have to find the volume by using the triple integral.

∫0π∫02π∫05p2sinϕdpdθdϕ
02

Evaluate integral

V=∫0π∫02π∫05p2sinϕdpdθdϕV=∫0πsinϕdϕ∫02πdθ∫05p2dpV=p3305∫0πsinϕdϕ∫02πdθV=1253∫0πsinϕdϕ∫02πdθV=1253(2π)∫0πsinϕdϕV=2503π-cosϕ0πV=2503π1+1V=5003πcubicunits

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Most popular questions from this chapter

Evaluate each of the double integrals in Exercises37-54as iterated integrals.

localid="1650380493598" ∫∫Rsin2x+ydA,

wherelocalid="1650380496793" R=x,y|0≤x≤π2and0≤y≤π2

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

∫∫RcosxydAwhereR=x,yIπ4≤x≤π2andπ2≤y≤π

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

f(x,y)=-2x2y3WhereR={(x,y)|-2≤x≤3,-1≤y≤5}

Use Definition 13.4to evaluate the double integrals in Exercises 29–32.

localid="1649936867482" ∬xx2ydA

whereR={(x,y)∣1≤x≤3and0≤y≤2}

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

∫∫Rx3ex2ydA,whereR={(x,y)|0≤x≤4and−1≤y≤1}
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