Q. 53 - Calculus

Question

The region between the graphs of $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=x^2$ on [0, 1], revolved around the x-axis 

Step-by-Step Solution

Answer

Volume is $\frac{\pi }{2} cubicunits$

Step 1: Given information

We are given two function $f\left(x\right)=\sqrt{x} and g\left(x\right)=x^2$

Step 2: Find the integral and evaluate it

We know that integral can be given as $V=2\pi {\int }_c^{d}r\left(y\right)h\left(y\right)dy$. As the axis of revolution is x-axis hence $r\left(y\right)=y$ and $h\left(y\right)=y^2$.

Substituting the values in the formula we get,

$$\begin{gathered} V=2\pi {\int }_0^{1}y\left(y^2\right)dy \\ V=2\pi {\int }_0^1{y}^{3}dy \\ V=2\pi {{\left[\frac{y^4}{4}\right]}^1}_{0}V=2\pi \left[\frac{1}{4}\right] \\ V=\frac{\pi }{2} \end{gathered}$$