Volume of the resulting solid?

Question

Consider the region in the plane enclosed by y = x^2 and y=x. We rotate this region about the line y=0. Find the volume of the resulting solid of revolution

Answer 1

The limits of integration are from zero to one.

Use the washer method.

Volume = ∫[πR² - πr²] dx = π∫[R² - r²] dx

= π∫[x² - (x²)²] dx = π∫[x² - x^4] dx

= π[(1/3)x³ - (1/5)x^5] | [Evaluated from 0 to 1]

= π[1/3 - 1/5] = 2π/15