Q. 31 - Calculus

Question

Consider the region between $f\left(x\right)=4-x^2$ and the x-axis on $[0, 2]$. For each line of rotation given in Exericses 29–32, use the shell method to construct definite integrals to find the volume of the resulting solid.

Step-by-Step Solution

Answer

The volume is $\frac{40}{3}\pi$. 

Step 1. Given Information.

We are given, 

Step 2. Finding the Volume.

As the region bounded by $f\left(x\right)=4-x^2$ and the x-axis from $x=0$, to $x=2$ is rotated around the line $x=2$, so to find the volume by shell method note that the radius will be $2-x$ and the height of the shell is given by $y=f\left(x\right)=4-x^2$

Therefore using the shell method

$$\begin{gathered} \text{ Volume }=2\pi {\int }_0^2(2-x)\left(4-x^2\right)dx \\ =2\pi {\int }_0^2\left(x^3-2x^2-4x+8\right)dx \\ =2\pi {\left[\frac{{\displaystyle 1}}{{\displaystyle 4}}{x}^4-\frac{{\displaystyle 2}}{{\displaystyle 3}}{x}^3-2x^2+8x\right]}_0^2 \\ =2\pi \left(\left[4-\frac{{\displaystyle 16}}{{\displaystyle 3}}-8+16\right]-\left[0\right]\right) \\ =\frac{{\displaystyle 40}}{{\displaystyle 3}}\pi \end{gathered}$$

Hence, the volume is $\frac{40}{3}\pi$.