Q. 26 - Calculus

Question

Write the volume of the two solids of revolution that follow in terms of definite integrals that represent accumulations of disks and/or washers. Do not compute the integrals.  

Step-by-Step Solution

Answer

The required volume is $\mathrm{\pi }{\int }_0^3\left({\left(\mathrm{f}\left(\mathrm{x}\right)\right)}^2-1^2\right)\mathrm{dx}$

Step 1. Given Information

The given figure is  

Step 2. Explanation

A solid of revolution is being formed by rotating the region around x-axis.

The given figure can be shown as below,

The part of region above x-axis is not bounded by x-axis. But it can be expressed as a washer with outer radius being the graph of the function and inner radius is the graph of $y=1$. The x-interval for this region is $\left[0,4\right]$

Thus, the integral of the volume of solid formed by rotating the given region is expressed as  $\mathrm{\pi }{\int }_0^3\left({\left(\mathrm{f}\left(\mathrm{x}\right)\right)}^2-1^2\right)\mathrm{dx}$