A solid has a base bounded above by the curve y=4−x2 and below by the x–axis for 0≤x≤2. Cross sections taken perpendicular to the x–axis are isosceles right triangles whose legs lie entirely in the base region.
(a) Express the length of a leg of the triangular cross section as a function of x. Then write an explicit formula for the cross–sectional area A(x).
(b) Set up, without evaluating, a definite integral that represents the volume V of the solid.
(c) Evaluate the integral in part (b) to determine the exact volume of the solid.