Suppose f is a continuous function defined on a rectangle R = [a, b] × [c, d].

(a) Write an expression for a double Riemann sum of f. If f(x, y) ≥ 0, what does the sum represent?

(b) Write the definition of ∫∫ _R f(x, y) dA as a limit.

(c) What is the geometric interpretation of ∫∫ _R f(x, y) dA if f(x, y) ≥ 0? What if f takes on both positive and negative values?

(d) How do you evaluate ∫∫ _R f(x, y) dA?

(e) What does the Midpoint Rule for double integrals say?

(f) Write an expression for the average value off.