Question

Find the volume of the solid obtained by rotating the region bounded by the curves `y=x^3` and `x`-axis in the interval `(1,2)`?

Answer 1

Volume is `18 1/7pi`

Explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves `y=x^3`, the `x`-axis and the lines `x=1` and `x=2` turn around the `x`-axis,

we need to find area of the curve under the curve `y=x^3`, between `x=1` and `x=2`.

As the same is rotated around `x`-axis, we will get the volume of the desired solid.

Hence this volume is `int_1^2pi(x^3)^2dx`

= `int_1^2pix^6dx`

= `pi[x^7/7]_1^2`

= `pi{2^7/7-1^7/7}=(127 pi)/7=18 1/7pi`