Q. 39 - Calculus

Question

Consider the region between $f\left(x\right)=4-x^2$ and the line y = 5 on [0, 2]. For each line of rotation given in Exercises 39–42, use the shell method to construct definite integrals to find the volume of the resulting solid.

Step-by-Step Solution

Answer

The integral can be given as $2\pi {\int }_0^{2}x(x^2+1)dx$ and the volume can be given as $4\pi$ cubicunits

Step 1: Given information

We are given a function $f\left(x\right)=4-x^2$ and

Step 2: Find the integral and evaluate it

We know that integral can be given as

$V=2\pi {\int }_a^{b}r\left(x\right)h\left(x\right)dx$ where r and h are radius and height respectively

The revolution is around y-axis hence the radius is $r\left(x\right)=x$

and the height can be given as

$h\left(x\right)=x^2+1$ and the interval is [0,2]

Substituting the values in integral can be given as

$$\begin{gathered} V=2\pi {\int }_0^2\left(x\right)(x^2+1)dx \\ V=2\pi {\int }_0^2{x}^3+x dx \\ V=2\pi {{[\frac{x^4}{4}+\frac{x^2}{2}]}^2}_0 \\ V=2\pi [4-2] \\ V=4\pi \end{gathered}$$