Q. 29 - Calculus

Question

Think about the area between the x-axis on $[0, 2]$ and $f\left(x\right)=4-x^2.$ Use the shell approach to create definite integrals for each line of rotation provided in this exercise to determine the volume of the resulting solid.

Step-by-Step Solution

Answer

The volume is $8\pi$.

Step 1: Given information.

Consider the given function,

$f\left(x\right)=4-x^2$

Step 2: Explanation.

When using the shell method to get the volume, keep in mind that the radius of the shell will be x and that the height of the shell is determined by $y=f\left(x\right)=4-x^2$ because the region bordered by $f\left(x\right)=4-x^2$ and the x-axis from $x=0$ to $x=2$ are rotated around the y-axis.

So utilizing the shell approach to find the volume.

$\begin{array}{rcl}\text{ Volume }& =& 2\pi {\int }_0^2\left(x\right)\left(4-x^2\right)dx\\ & =& 2\pi {\int }_0^2\left(4x-x^3\right)dx\\ & =& 2\pi {\left[\frac{4}{2}{x}^2-\frac{1}{4}{x}^4\right]}_0^2\text{ upon integration }\\ & =& 2\pi \left(\right[8-4]-[0-0\left]\right)\text{ simplifying }\end{array}$

Therefore the volume is$8\pi$.