Question

The region R is bounded by the curves y = 22 +1 and y = 3x - 1. Set up an integral blume of the solid obtained by rotating R about the line x = 4. Then calculate the
calculate volume

Answer 1

we are given equations as

$y=x^2+1$

$y=3x-1$

It is rotated about x=4 line

Firstly, we can draw graph

30 x=4 20 y=r' +1 -10 (2,5) (1,2) y = 3x - 1 2 3 5 6

Bound is from 1 to 2

we can use shell method

V = 2arhda

$h=(3x-1)-(x^2+1)$

$r=4-x$

now, we can set up integral

V = 27(4 - 0) (3x – 1) - (x2 + 1))d.

$=2\pi \cdot \int _1^2x^3-7x^2+14x-8dx$

$=2\pi \left(\int _1^2x^3dx-\int _1^27x^2dx+\int _1^214xdx-\int _1^28dx\right)$

we can solve each integrals

and then combine them

$=2\pi \left(\frac{15}{4}-\frac{49}{3}+21-8\right)$

$V=\frac{5\pi }{6}$............Answer