Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

Question

y = 1/sprt(x), y = 0, x = 1, x = 8
a. π/8 b. (π ln 8)/2 c.π * sprt(8) d. π*ln 8

Answer 1

The area of the curve in the x - y plane is ∫1/√x dx evaluated between x = 1 and x = 8. The volume is the area times the length of the centroid rotated the axis specified (which wasn't).
A = ∫1/√x dx = 2√x [1 - 8] = 2√8 - 2.

x coordinate of centroid is ∫ x*y dx / ∫ y dx = ∫ x/√x dx / A = (3/2)x^3 [8 - 1] / A

xc = [8^3/2 - 1] / A

The y coordinate is yc = 1/√xc.

The volume rotated around the y axis is

2*π*xc*A = 2*π*(8^3/2 - 1)

Answer 2

Assume it is revolving about x-axis.

v
= ∫πy^2 dx, x from 1 to 8
= ∫π(1/x)dx, x from 1 to 8
= π*ln8