Q. 13 - Calculus

Question

In Exercises 13–20, we explore the relationship between the

shell method for finding volumes of solids of revolution dis-

cussed in Chapter 6 and the method of double integrals using

polar coordinates.

Sketch a function $z = f \left(x\right)$ in the xz-plane such that $f \left(x\right) \ge 0$ on the interval $[0, b].$ Use the shell method to find an integral that represents the volume of the solid of revolution obtained when the region bounded above by the graph of $f$ and bounded below by the x-axis on the interval $[0, b]$

is rotated about the z-axis.

Step-by-Step Solution

Answer

The required volume generated is found to be $V={\int }_0^{b}2\pi xf\left(x\right)dx$

Step 1 Given Information

The objective of this problem is to sketch a function $z=f\left(x\right)$ in $x-z$ plane such that $f\left(x\right)\ge 0$ on the interval $[0, b]$. 

Step 2 Calculation

 Use shell method to find an integral that represents the volume of the solid of revolution obtained when the region bounded above by the graph of $f$ and bounded below by $x$-axis on the interval $[0,b]$ is rotated about the $z$-axis.

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

$V={\int }_0^{b}2\pi xzdx$

$V={\int }_0^{b}2\pi xf\left(x\right)dx$

Thus, the volume of the solid generated is

$V={\int }_0^{b}2\pi xf\left(x\right)dx$