Q. 43 - Calculus

Question

Consider the region between f(x) = x+1 and the line y = −2 on [1, 5]. For each line of rotation given in Exercises 43 and 44, 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 $2\pi {\int }_1^5(5-x)(x+3)dx$ and the value of integral is $\frac{256}{3}\pi cubicunits$

Step 1: Given information

We are given a function f(x)= x+1 and

Step 2: Find the integral and evaluate it

We know that integral can be given as $V=2\pi {\int }_a^{b}r\left(x\right)h\left(x\right)dx$ as the axis of revolution is x=5

Hence the radius can be given as $r\left(x\right)=5-x$ and the height can be given as $h\left(x\right)=x+3$

Substituting the values we get,

$$\begin{gathered} V=2\pi {\int }_1^5(5-x)(x+3)dx \\ V=2\pi {\int }_1^5(5x+15-x^2-3x)dx \\ V=2\pi {{[x^2+15x-\frac{x^3}{3}]}^5}_{1}V=\frac{256}{3}\pi \end{gathered}$$