e2 log(x) Example Consider the integral x(e2y – 2e4y)10 dy dx = 1 Observe the difficulty with evaluating the integral using the given order of integration. Reverse the order of integration, and evaluate the integral. y (e2 , 2) y = log(x) 2 e2 x(e2y – 2e4y)10 dx dy = x (1,0) (e2,0) ey e2 2 2 x2(e2y – 2e4y)10 ————— dy = 2 (e4 – e2y)(e2y – 2e4y)10 ————————— dy = 2 x = ey

2 2 (e4 – e2y)(e2y – 2e4y)10 ————————— dy = 2 – (e2y – 2e4y)11 —————— = 44 y = 0 1 – (e4 – 4e4)11 —————— = 44 1 + 311e44 ———— . 44

Look at the Mean Value Theorem for Double Integrals on page 352 of the textbook, and use the theorem to find bounds on the integral 1 ———— dx dy where D is the unit circle of radius 1/8 1 + x2 + y2 centered at the origin. D 1 Since  ————  , then 1 + x2 + y2 64 — 65 1 1 ———— dx dy 1 + x2 + y2  — 65  — 64   D