Which of the following integrals determines the volume of a cone of height H and radius R?
A) $\int_{0}^{R} \pi(R(H-h))^{2} d h$
B) $\int_{0}^{R} \pi\left(\frac{H}{R}(H-h)\right)^{2} d h$
C) $\int_{0}^{H} \pi\left(\frac{R}{H}(H-h)\right)^{2} d h$
D) $\int_{0}^{H} \pi(R(H-h))^{2} d h$

Question

Quizplus · Accepted Answer

The volume of a cone can be found using the integral of the areas of circles with radii that change linearly from the base to the apex as you move along the height. The correct formula that represents this is $\int_{0}^{H} \pi\left(\frac{R}{H}(H-h)\right)^{2} d h$, where $h$ varies from 0 to $H$, and the radius of the circles changes linearly with $h$. This is accurately represented in option C, where the radius at a height $h$ is given by $\frac{R}{H}(H-h)$, ensuring the radius decreases linearly from $R$ at the base to 0 at the apex.

Which of the Following Integrals Determines the Volume of a Cone