find the volume of the solid obtained by the rotating the region bounded by the given curves about the ...?

Question

find the volume of the solid obtained by the rotating the region bounded by the given curves about the specified axis:
y=1/x^3
y=0
x=3
x=7
about y=-4

Answer 1

I'm just taking a shot at it... No guarantees
7 1/x^3+4 2pi
∫ ∫ ∫ 1 dz dy dx
3 4 0

7 1/x^3+4
∫ ∫ 2pi dy dx
3 4

7
∫ 1/x^3+4 - 4 dx * 2pi
3

7
∫ 1/x^3 dx * 2pi
3

2pi * -1/2 x^(-2) |7 to 3 = -pi(1/49 - 1/9) ≈ .28495

Answer 2

I would use the method of washers.

The outer radius R is (x^(-3) - (-4)) = (x^(-3) + 4)
The inner radius r is (0 - (-4)) = 4

dV = π(R² - r²)dx = π [(x^(-3) + 4)² - 16] dx

V = integral from 3 to 7 of π [(x^(-3) + 4)² - 16] dx