Find the volume of the solid obtained by rotating the region bounded by y=x and y=sqrt(x) about the line x=4.

Answer 1

The general formula for washers is like π r², where you need to find an inner and outer radius:

π ∫ (outter²-inner²) dx/y

y=sqrt(x) => x=y^2, and x=y

π ∫ [(4 - y²)² - (4 - y)²] dy (integrate from 0 to 1)

π ∫ [16 - 8y² +y⁴ - (16 - 8y + y²)] dy expand

π ∫ [y⁴ - 9y² + 8y] dy simplify by collecting like terms

π [y⁵/5 - 3y³ + 4y] from 0 to 1 integrate using power rule

π [1/5 - 3 + 4] = 6π/5 sub upper limit - lower limit and simplify

Drawing a picture helps. :)