Q. 46 - Calculus

Question

Consider the region between and g(x) = x on [1, 4]. For each line of rotation given in Exercises 45–48, 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 $V=2\pi {\int }_a^{b}x({(x-2)}^2-x)dx$ and the value of integral is $190.77\pi$.

Step 1: Given information

We are given functions $f\left(x\right)={(x-2)}^2$ and g(x) =  x

Also

Step 2: Find the integral and evaluate it

We know that integral can be given as $V=2\pi {\int }_a^{b}r\left(x\right)h\left(x\right)dx$ as the axis of rotation is y-axis the radius is r(x) = x and the height can be given as $h\left(x\right)=({(x-2)}^2-x)$

Substituting the values in the integral we get,

$$\begin{gathered} V=2\pi {\int }_a^{b}x({(x-2)}^2-x)dx \\ V=2\pi {\int }_1^{4}x(x^2-4x+4-x)dx \\ V=2\pi {\int }_1^4(x^3-5x^2+4x)dx \\ V=2\pi {{[\frac{x^4}{4}-\frac{5}{3}{x}^3+2x^2]}^4}_1 \\ V=2\pi [95.38] \\ V=190.77\pi \end{gathered}$$