Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

y = 2/x, y = 0, x = 1, x = 3; about y = −1

V=

Answer 1

To find the volume, we use the method of cylindrical shells, which gives us the formula:

V = ∫[a,b] 2πrh dx
where r is the radius and h is the height.

In this case:
- The region is bounded by y = 2/x, y = 0, x = 1, and x = 3.
- We have to rotate it around y = -1.
- So, the radius r = y+1 = 2/x + 1 and the height h = x.

Now, we can plug the values into the formula and integrate:

V = ∫[1,3] 2π(2/x + 1)x dx
= 2π ∫[1,3] (x(2/x) + x) dx
= 2π ∫[1,3] (2 + x) dx

Now, we integrate:

V = 2π [2x + (1/2)x^2] evaluated from 1 to 3
= 2π [(2(3) + (1/2)(3^2)) - (2(1) + (1/2)(1^2))]
= 2π [(6 + 4.5) - (2 + 0.5)]
= 2π (10.5 - 2.5)
= 2π (8)

V = 16π cubic units.