Question

Rotate the region bounded by x

=

(

y

−

2

)

2

x=(y−2)2

, the x

x

-axis and the y

y

-axis about the x

x

-axis.

Answer 1

Given, the bounded region

Volume generated by the bounded region about the x-axis=$2 \pi \int_{a}^{b}y^2(x)dx\space where \space x \space varies \space from \space a\space to\space b.\newline =2 \pi \int_{0}^{4}(2- \sqrt{x})^2dx\newline =2 \pi \int_{0}^{4}(4+x-4 \sqrt{x})dx\newline =2 \pi [4x+\frac{x^2}{2}-8 \frac{x^{\frac{3}{2}}}{3}]_{0}^{4}\newline =2\pi (\frac{8}{3})\newline =\frac{16}{3} \pi\space units^3$

Thus, the required volume is $\frac{16}{3} \pi\space units^3$.