Question

Consider the region between $f\left(x\right)=\sqrt{x}$and the x-axis on $[0,4]$. For each line of rotation given in Exercises 27–30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

Short answer

The required volume is $\frac{583}{16}\mathrm{\pi }$

Step-by-step solution

Step 1. Given Information  

The given figure is

Step 2. Calculation  

To determine, the volume of solid of revolution rotated around vertical lines express the curve as inverse function.

$$\begin{gathered} f\left(x\right)=\sqrt{x} \\ y=\sqrt{x} \\ {y}^2=x \\ x=y^2 \\ g\left(y\right)=y^2 \end{gathered}$$

For the x-interval of $[0,4]$, the corresponding interval of y-variable will be $[0,2]$

The width of each washer is determined as,

$$\begin{gathered} ∆y=\frac{b-a}{n} \\ =\frac{2-0}{4} \\ =\frac{1}{2} \end{gathered}$$

The external radius is $5-g\left(y_k\right)$and inner radius is given as 1.

The starting value of disk is 0. It can be termed as $y_0$.

Determine these end points as follows,

$$\begin{gathered} y_k=y_0+k∆y \\ =0+k\left(\frac{1}{2}\right) \\ =\frac{k}{2} \end{gathered}$$

Step 3. Calculation  

Total volume of solid is determined as follows,

$$\begin{gathered} V=\underset{k=1}{\overset{n}{\sum \mathrm{\pi }}}\left({\left(R\left(y\right)\right)}^2-{\left(r\left(y\right)\right)}^2\right)∆y \\ =\underset{\mathrm{k}=1}{\overset{\mathrm{n}}{\sum \mathrm{\pi }}}\left({\left(5-\mathrm{g}\left({\mathrm{y}}_{\mathrm{k}}\right)\right)}^2-1^2\right)\frac{1}{2} \\ =\frac{\mathrm{\pi }}{2}\sum _{k=1}^n\left({\left(5-{y_k}^2\right)}^2-1^2\right) \\ =\frac{\mathrm{\pi }}{2}\sum _{k=1}^4\left({\left(5-\frac{k^2}{4}\right)}^2-1\right) \\ =\frac{\mathrm{\pi }}{2}\left(\left({\left(5-\frac{1^2}{4}\right)}^2-1\right)+\left({\left(5-\frac{2^2}{4}\right)}^2-1\right)+\left({\left(5-\frac{3^2}{4}\right)}^2-1\right)+\left({\left(5-\frac{4^2}{4}\right)}^2-1\right)\right) \\ =\frac{\mathrm{\pi }}{2}\left(\frac{345}{16}+\frac{308}{16}+\frac{273}{16}+15\right) \\ =\frac{583}{16}\mathrm{\pi } \end{gathered}$$