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Chapter 13: Q. 16 (page 1027)

Use the techniques of this section to obtain an iterated integral that employs polar coordinates to represent the volume of the solid discussed in Exercise 15.

Short Answer

Expert verified

The volume of the solid generated isV=∫0r-t2πrcos2θf(r)dr

Step by step solution

01

Given Information

The objective of this problem is to represents the volume of the solid of revolution obtained when the region bounded above by the graph of fand bounded below by x-axis on the interval [0,b]is rotated about the z-axis.

02

Calculation

Use shell method for the volume generated by the region bounded by the graph.

V=∫0b2πxzdx

V=∫0b2πxg(x,y)dx

V=∫0b2πxfx2+y2dx

Substitute x=rcosθ,y=rsinθ

V=∫0r-b2πrcosθfr2cos2θ+r2sin2θcosθdr

V=∫0r-b2πrcos2θf(r)dr

Thus, the volume of the solid generated is

V=∫0r-t2πrcos2θf(r)dr

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