The region R is bounded by the x-axis, y-axis, x = 3 and y = 1/(sqrt(x+1))

Question

A. Find the area of region R.
B. Find the volume of the solid formed when the region R is revolved about the x-axis.
C. The solid formed in part B is divided into two solids of equal volume by a plane perpendicular to the
x-axis. Find the x-value where this plane intersects the x-axis.

Please show your work!

Answer 1

A.
∫[0,3] 1/√(x+1) dx
= 2√(x+1) [0,3]
= 2

B. using discs, v=∫πy^2 dx
= π∫[0,3] 1/(x+1) dx
= πlog(x+1) [0,3]
= πlog4

C. We want c where
π∫[0,c] 1/(x+1) dx = π∫[c,3] 1/(x+1) dx
log(c+1)-log(0+1) = log(3+1)-log(c+1)
2log(c+1) = log(4)
log(c+1) = log(2) (and not because 4/2=2)
c = 1