Chapter 15.1, Problem 38E

Consider the solid region $S$ that lies under the surface $z=x^2\sqrt{y}$ and above the rectangle $R=[0,2]\times [1,4]$

(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.

(c) Find the value of S.