Question

Use the method disks/rings to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis of y=2x+1, x=4 and y= 3 about the line x=-4.

Answer 1

The given lines are represented as-

The region formed by rotating the above regions is-

So, the inner and outer radii are as follows:

Inner Radius $r=4+\dfrac{1}{2}(y-1)=\dfrac{y}{2}+\dfrac{7}{2}$

Outer radius $R=4+4=8$

Area of the ring is

$A(x)=\pi [R^2-r^2]$

   $=\pi[(8)^2-(\dfrac{y}{2}+\dfrac{7}{2})^2]$

   $=\pi(\dfrac{207}{4}-\dfrac{7}{2}y-\dfrac{y^2}{4})$

Then the volume of the solid obtained is

 $V=\int_3^9A(x)dy=\pi(\dfrac{207}{4}-\dfrac{7}{2}y-\dfrac{y^2}{4})=\int_3^9 \pi(\dfrac{207}{4}-\dfrac{7}{2}y-\dfrac{y^2}{4})dy$

  $=\pi(\dfrac{207y}{4}-\dfrac{7y^2}{4}-\dfrac{y^3}{12})|_3^9=126\pi$