Applications of Integration

Question

How would you prove the volume of a cone equation by rotating the line about the x-axis if you have (h,r) and (h,0)? Start by finding the equation of the line.

Answer 1

Use the Disk Method to solve the volume:

π ∫_a^b [f(x)]² dx, where a ≤ x ≤ b

Assuming r and h are constant and f(x) = (r/h)x, then our integral will be:

V= π ∫₀^h [(r/h)x]² dx, where 0 ≤ x ≤ h.

π ∫₀^h [(r²/h²)x²] dx

=(1/3)π [(r²/h²)x³]₀^h

=(1/3)π [(r²/h²)h³ - 0

= (1/3)π ⋅r²⋅h