Some concepts regarding area that I understanding

[fwb]

Most 3d volume formulas look similar to their corresponding 2d area formulas for a particular shape. How does calculus, specifically integration, relate the two sets of formulas?

For example: how is the formula of a cone derived from the formula of a circle?

 

[mathman]

One method: 3d coordinate system - put cone point at origin and cone axis along positive x axis. Let h be height of cone and R=radius of base. Let k=R/h. The radius of the cone circle defined by a plane at point x perpendicular to the x-axis is given by r=kx. The differential of volume at a point x is then pi*r²dx. Integrate from 0 to h and the volume will be pi*k²h³/3=pi*R²h/3.

There may be neater ways, but you can see the point.