Use the Shell Method to find the volume of a solid obtained by rotating the region B about the x-axis.

Question

y=(x^2)+b

Assume a=2 and b=3.

Graph link:

https://drive.google.com/file/d/14d6YUX7OliWwHJl7nddFR5GLsMZYHJqU/view?usp=sharing

Answer 1

The easiest way to do this, by the shell method is to find the rotated volume of A and B and subtract the rotated volume of A.

V_B = V_AB - V_A

The V_AB = the volume of a cylinder of radius 2²+3=7 and length 2. V_AB = πr²h =98π

The rotated volume of A by shell method is:

V_A = 2π∫⁷₃ yf(y)dy = 2π∫⁷₃ y√(y-3) dy

Let u = y-3

du = dy

y = u+3

when y = 3, u=0

when y=7, u=4

V_A = 2π∫⁴₀(u+3)√u du = 2π∫⁴₀ u^3/2+3u^1/2du = 2π[2u^5/2/5 + 3(2/3)u^3/2]⁴₀ = 288π/5

V_B = 98π - 288π/5 = 202π/5

If you use the disk method to get the rotated volume of region B, it comes out the same.