Q. 22 - Calculus

Question

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.   

$\mathrm{\pi }{\int }_0^2\mathrm{ydy}$

Step-by-Step Solution

Answer

The required region is shown as,   

Step 1. Given Information

The given integral is $\mathrm{\pi }{\int }_0^2\mathrm{ydy}$

Step 2. Explanation

The given integral is of the form $\mathrm{\pi }{\int }_a^b(\mathrm{f}{\left(\mathrm{y}\right))}^2\mathrm{dy}$ which represents the volume of a solid formed by rotating the region bound by f(y) and y-axis in the interval $[a,b]$ around y-axis.

On comparing the given region with the above region, we have,

$f\left(y\right)=\sqrt{y} \mathrm{in} \mathrm{the} \mathrm{interval} [0,2]$

Express f(y) as x and rewrite the function,

$$\begin{gathered} x=\sqrt{y} \\ {x}^2=y \\ y=x^2 \end{gathered}$$

The simplified function represents the parabola.

Plot the graph of the function and identify the region between the intervals.