Q. 27 - Calculus

Question

Consider the region between $f\left(x\right)=\sqrt{x}$and the x-axis on $\left[0,4\right]$. 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 $10\mathrm{\pi }$

Step 1. Given Information

The given figure is   

Step 2. Calculation

The width of each disk is determined as,

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

The radius of each disk is given as $f\left(x_k\right)$. The starting value of disk is 0. It can be termed as $x_0$.

Determine these end points as follows,

$$\begin{gathered} x_k=x_0+k∆x \\ =0+k∆x \\ =k∆x \end{gathered}$$

Step 3. Calculation

Use different values of k to determine the values of $x_k$

$$\begin{gathered} x_1=1\left(1\right)=1 \\ {x}_2=2\left(1\right)=2 \\ {x}_3=3\left(1\right)=3 \\ {x}_4=4\left(1\right)=4 \end{gathered}$$

Total volume of solid is determined as follows,

  $$\begin{gathered} V=\mathrm{\pi }\sum _{\mathrm{k}=1}^{\mathrm{n}}{\left(\mathrm{f}\left({\mathrm{x}}_{\mathrm{k}}\right)\right)}^2∆\mathrm{x} \\ =\mathrm{\pi }\sum _{\mathrm{k}=1}^4{\left(\sqrt{{\mathrm{x}}_{\mathrm{k}}}\right)}^2\left(1\right) \\ =\mathrm{\pi }\left({\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3+{\mathrm{x}}_4\right) \\ =\mathrm{\pi }(1+2+3+4) \\ =10\mathrm{\pi } \end{gathered}$$