volume of a solid

Question

Find the volume of the solid obtained by revolving the graph of the line y=x on the interval [0,h] around the x axis

Answer 1

Well we can get there without using calculus if we wish: this 3-d solid is a cone of height h and with radius h.

From geometry, volume of a cone = 1/3πr²h = 1/3πh³

Throwing calculus at it has us doing this volume of rotation by disks as follows:

V = ∫₀^h πx² dx = 1/3πx³ ]₀^h = 1/3πh³