9. Let R = [a, b] x [c, d] be a two-dimensional rectangle and f : R ---- R be bounded.

(a) Prove that

(L) fLfdA ~ (L) lb ((X) ld f(X,Y)dY) dx

~ (U) lb ((X) ld f(x, y) dY) dx

~ (U) fLfdA for X = U or X = L.

(b) Prove that if f is integrable on R, then flfdA~ l ((L) t f(x, Y) dY) dx~ l ((u) l' f(x, Y) dY) dx.

(c) Compute the two iterated integrals in part

(b) for f(x,y) = { ~ yE Q y~ Q and R = [0,1] x [0,1]. Prove that f is not integrable on R.