Q. 4 - Calculus

Question

Write down definite integrals to express the given geometric quantities :

The volume of the solid is obtained by revolving $f\left(x\right)$ on $\left[a,b\right]$ around the y-axis, by the shell method. 

Step-by-Step Solution

Answer

The volume of solid with nested shell along y-axis $=2\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{h}\left(\mathrm{x}\right) d\mathrm{y}$

Step 1: Given Information :

Given that to assume that $f\left(x\right)$ is continuous and differentiable, with a continuous derivative. 

Step 2: Definite integral formula by shell method :

The volume of the solid S formed by revolving the region bounded by the curve $y=f\left(x\right)$ between $y=a$ and $y=b$ about the y−axis is given by,

$V=2\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{h}\left(\mathrm{x}\right) d\mathrm{y}$

Where,  $f\left(x\right)$ and $h\left(x\right)$  are the radius and height functions for the shells in terms of x.