Q. 41 - 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 $2\pi {\int }_0^2(3x^2+3-x^3-x)dx$ and its value is $16\pi$

Step 1: Given information

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

Step 2: Find the integral and evaluate it

We know that integral can be given as

$V=2\pi {\int }_0^{2}r\left(x\right)h\left(x\right)dx$where r and h are the radius and height respectively

the axis of revolution is x=3 hence the radius is $r\left(x\right)=3-x$ and height can be given as $h\left(x\right)=x^2+1$ and interval is [0,2]

Substituting the values we get

$$\begin{gathered} V=2\pi {\int }_0^2(3-x)(x^2+1)dx \\ V=2\pi {\int }_0^2(3x^2+3-x^3-x)dx \\ V=2\pi {{[x^3+3x-\frac{x^4}{4}-\frac{x^2}{2}]}^2}_0 \\ V=16\pi cubicunits \end{gathered}$$