Q. 3 - Calculus

Question

Consider the rectangle bounded by y = 3 and y = 0 on the x-interval $[2,2.5]$.

(a) What is the volume of the disk obtained by rotating this rectangle around the x-axis?

(b) What is the volume of the washer obtained by rotating this rectangle around the line y=5?

Step-by-Step Solution

Answer

(a) The volume of the disk obtained by rotating this rectangle around the x-axis is $2.25\mathrm{\pi }$.

(b) The volume of the washer obtained by rotating this rectangle around the line $y=5$ is $5.25\mathrm{\pi }$.

Part (a) Step 1. Given Information.

The rectangle bounded by y = 3 and y = 0  on $[2,2.5]$

Part (a) Step 2. Volume of the disk.

The volume of the disk formed by rotating around the x-axis is given by,

$V={\int }_a^b{\left[r\right(x\left)\right]}^{2}dx$ where r(x) is the radius of the disk.

From the rectangle, $r\left(x\right)=3$

Part (a) Step 3. Volume of the disk.

The volume of the disk is determined by,

$$\begin{gathered} V=\mathrm{\pi }{\int }_2^{2.5}{3}^2\mathrm{dx} \\ =9\mathrm{\pi }{\int }_2^{2.25}\mathrm{dx} \\ =9\mathrm{\pi }{\left[\mathrm{x}\right]}_2^{2.25} \\ =9\mathrm{\pi }[2.25-2] \\ =2.25\mathrm{\pi } \end{gathered}$$

Part (b) Step 1. Draw the region.

Draw the region along with the rotation on the coordinate plane.

Part (b) Step 2. Volume by Washer method.

The volume of washer formed is given by the formula,

$V={\int }_a^b\left[\right[R{\left(x\right))}^2-\left(r{\left(x\right))}^2\right]dx$

$R\left(x\right)=5; r\left(x\right)=2$

Part (b) Step 3. Find the volume.

The volume is,

$$\begin{gathered} V=\mathrm{\pi }{\int }_2^{2.25}{5}^2-2^{2}dx \\ =21\mathrm{\pi }{\int }_2^{2.25}\mathrm{dx} \\ =21\mathrm{\pi }{\left[\mathrm{x}\right]}_2^{2.25} \\ =5.25\mathrm{\pi } \end{gathered}$$