Consider the solid region S that lies under the surface z = x 2 y and above

Question:

Consider the solid region S that lies under the surface z = x²√y and above the rectangle R = [0, 2] c [1, 4].

(a) Find a formula for the area of a cross-section of S in the plane perpendicular to the x-axis at x for 0 ≤ x ≤ 2. Then use the formula to compute the areas of the cross-sections illustrated.

(b) Find a formula for the area of a cross-section of S in the plane perpendicular to the y-axis at y for 1 ≤ y ≤ 4. Then use the formula to compute the areas of the cross-sections illustrated.