Q. 18 - Calculus

Question

Write the volumes of the solids of revolution shown in Exercises 17–20 in terms of definite integrals that represent accumulations of shells. Do not solve the integrals.

Step-by-Step Solution

Answer

The volume of the solid in terms of definite integral is $2\mathrm{\pi }{\int }_0^{2}xf\left(x\right)dx$.

Step 1. Given information.

Consider the given figure that represent accumulations of shells. 

Step 2. Find the volume of solid in terms of definite integral.

The curve is rotated about the y-axis. So, the function in terms of x is $y=f\left(x\right)$.

Accumulating the shells from the inside out, from $x=0$ at the center to $x=2$ outside, and apply the function $y=f\left(x\right)$.

The volume of the solid in terms of definite integral is:

$V=2\mathrm{\pi }{\int }_0^{2}xf\left(x\right)dx$