Question

What will be the volume of the solid of revolution formed by rotating the finite region bounded by the graphs of $y=\sqrt{x}$ and $y=x^3$ about the x-axis?

• A. $\frac{5\pi }{3}$
• B. $\frac{5\pi }{14}$
• C. $\frac{5\pi }{7}$
• D. $\frac{7\pi }{5}$

Answer 1

Answer: (B)

$\frac{5\pi }{14}$

Outer radius$=x^{1/2}$

Inner radius$=x^3$

Hence

$V=\pi {\int }_0^1\left[\right(x^{1/2}{)}^2-\left(x^3{)}^2\right]dx$

$=\pi {\int }_0^1[x-x^6]dx$

$=\pi {[\frac{x^2}{2}-\frac{x^7}{7}]}_0^1$

$=\pi \left[\frac{7-2}{14}\right]$

$=\frac{5\pi }{14}$