The region R is bounded by the x-axis, x = 1, x = 3, and y = 1/x^3

Question

A) Find the area of R

B) B. Find the value of h, such that the vertical line x = h divides the region R into two Regions of equal area.

Answer 1

A = ∫[1,3] 1/x^3 dx
= -1/2x^2 [1,3]
= 4/9

Now, we want to find c such that

∫[1,c] 1/x^3 dx = ∫[c,3] 1/x^3 dx
(-1/2c^2 - (-1/2)) = (-1/18 - (-1/2c^2))
(c^2-1)/2c^2 = (9-c^2)/18c^2
c = 3/√5

Answer 2

Thank you!!

Answer 3

C. Find the volume of the solid generated when R is revolved about the x-axis.

D. The vertical line x = k divides the region R into two regions such that when these two regions are revolved about the x-axis, they generate solids with equal volumes. Find the value of k.