Consider a solid shape whose base is defined as the ordinate set of a function on an interval . The cross sections of this solid taken perpendicular to the interval are squares. The volume of the solid is given by the formula
If is continuous on calculate the value .
We are given the expression
However, we also know we can compute the volume as the integral from 0 to of the area of each cross section. Since the cross sections are squares and the length of the edge on the base is given by (since the base is the ordinate set of , the length of the base of a cross section at a point is ), we know the area of each cross section is . Therefore, we have another expression for the volume of the solid given by
Setting these two expressions for the volume equal and differentiating both sides we have