Applications of Integration

Question

Find, by slicing, the volume of a cone whose height is 3 cm and whose base radius is 3 cm.

volume =

Answer 1

Let y = the height of the cone and r = the radius of each circle slice.

By slicing

Volume, V = ∫_a^b Area dy

The Area will be a circle area, which is πr². The relationship between the height and the radius is found by similar triangles, r/y = 3/3 or r = y. For this case we will invert the cone so that we go from zero radius to maximum radius.

V = π∫₀³r²dy = π∫₀³y²dy = πy³/3 |₀³ = 27/3π = 9π