Q. 30 - Calculus

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.

Step-by-Step Solution

Answer

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

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,  

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