Question

Show that the integrals from Exercises 18 and 19 evaluate to the same quantity.

Short answer

The solid of revolution, the shell method and the method of polar coordinates represent the same volume generated by the revolution of curve $z=f\left(x\right)$ about $z$ - axis.

Step-by-step solution

Given Information

The objective of this problem is to show that problems 18 to 19 represent the volume of the solid of revolution obtained when the region bounded above by the graph of $f$ and bounded below by $x$- axis on the interval $[a,b]$is rotated about the $z$-axis.

Calculation

The solid of revolution, the shell method and the method of polar coordinates represent the same volume generated by the revolution of curve $z=f\left(x\right)$about$z$-axis.