   Chapter 15.6, Problem 20E

Chapter
Section
Textbook Problem

Use a triple integral to find the volume of the given solid.20. The solid enclosed by the paraboloids y = x2 + z2 and y = 8 - x2 - z2

To determine

To find: The volume of the solid enclosed by the paraboloids y=x2+z2 and y=8x2z2 by using the triple integral.

Explanation

Paraboloids intersect each other to form solid, to find the intersection equate the given surfaces.

x2+z2=8x2z22x2+2z2=82(x2+z2)=8x2+z2=4

The above equation in polar form so, by using polar coordinates let x=rcosθ and z=rsinθ.

The paraboloids in circle form so take the value of θ is from 0 to 2π and the value of r is from 0 to 2.

Substitute x=rcosθ and z=rsinθ in given surfaces to get y varies from r2 to 8r2.

Let E be the region enclosed by solid where E={(θ,r,y)|0θ2π,0r2,r2y8r2}.

The volume of integral for the solid is, EdV=02π02r28r2dyrdrdθ.

Integrate the given integral with respect to y and apply the limit of it.

EdV=02π02[y]r28r2rdrdθ=02π02[8r2r

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