5.5 Areas of sections

For future use we note that $\int_{a}^{b}f(x,y)dx$ equals the area of the section that is cut by the plane through $(0,y,0)$ parallel to the co–ordinate plane $0xz$; whereas $\int_{c}^{d}f(x,y)dy$ equals the area of the section that is cut by the plane through $(x,0,0)$ parallel to the co–ordinate plane $0yz$. (Think of a loaf of bread, and cutting the bread into slices.)

Solution. We start by sketching the regions.