Question

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs:

y =

2 + x

x = 2
x = 8

Find the volume generated by revolving about the x-axis the regions bounded by the following graphs.

y = ex
x = -3
x = 1

Answer 1

For the first problem, we can use the disk method to find the volume generated by revolving the region bounded by y = 2 + x, x = 2, and x = 8 about the x-axis. The radius of each disk is y - 2, and the thickness is dx, so the volume is:

V = ∫2^8 π(y - 2)^2 dx

= ∫2^8 π(x^2 + 4x + 4 - 4x - 4) dx

= ∫2^8 π(x^2 + 4x - 4) dx

= π[(1/3)x^3 + 2x^2 - 4x]2^8

= 248π

For the second problem, we can use the washer method to find the volume generated by revolving the region bounded by y = ex, x = -3, and x = 1 about the x-axis. The outer radius of each washer is e^x, the inner radius is 0, and the thickness is dx, so the volume is:

V = ∫-3^1 π(e^x)^2 dx

= ∫-3^1 πe^(2x) dx

= (1/2)π(e^2x)|-3^1

= (e^2 - e^-6)π/2