Chapter 14.5, Problem 32E

HOW DO YOU SEE IT? Consider the surface $f(x,y)=x^2+y^2$ (see figure) and the surface area $f$ that lies above each region $R$. Without integrating, order the surface areas from least to greatest. Explain.

(a) $R:\text{rectangle with vertices }(0,0),(2,0),(2,2),(0,2)$

(b) $R:\text{triangle with vertices }(0,0),(2,0),(2,2)$

(c) $R=\left\{(x,y):x^2+y^2\le 4,\text{first quadrant only}\right\}$