Q. 29 - 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 $\mathrm{\pi }\left(14+2\sqrt{2}+2\sqrt{3}\right)$

Step 1. Given Information

The given figure is     

Step 2. Calculation

The width of each washer is determined as,

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

The external radius is $1+f\left(x_k\right)$ 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,

$$\begin{gathered} x_k=x_0+k∆x \\ =0+k\left(1\right) \\ =k \end{gathered}$$

Step 3. Calculation

Total volume of solid is determined as follows,  

$$\begin{gathered} V=\sum _{k=1}^n\mathrm{\pi }\left({\left(\mathrm{R}\left(\mathrm{x}\right)\right)}^2-{\left(\mathrm{r}\left(\mathrm{x}\right)\right)}^2\right)∆\mathrm{x} \\ =\mathrm{\pi }\sum _{\mathrm{k}=1}^4\left({\left(1+\mathrm{f}\left({\mathrm{x}}_{\mathrm{k}}\right)\right)}^2-1^2\right)\left(1\right) \\ =\mathrm{\pi }\sum _{\mathrm{k}=1}^4\left({\left(1+\sqrt{{\mathrm{x}}_{\mathrm{k}}}\right)}^2-1\right) \\ =\mathrm{\pi }\sum _{\mathrm{k}=1}^4\left({\left(1+\sqrt{\mathrm{k}}\right)}^2-1\right) \\ =\mathrm{\pi }\left(\left({\left(1+\sqrt{1}\right)}^2-1\right)+\left({\left(1+\sqrt{2}\right)}^2-1\right)+\left({\left(1+\sqrt{3}\right)}^2-1\right)+\left({\left(1+\sqrt{4}\right)}^2-1\right)\right) \\ =\mathrm{\pi }\left(3+\left(2+2\sqrt{2}\right)+\left(3+2\sqrt{3}\right)+8\right) \\ =\mathrm{\pi }\left(14+2\sqrt{2}+2\sqrt{3}\right) \end{gathered}$$