Question

2. (10 points) In prior Calculus courses, we made use of limits to calculate improper integrals of the form of.dr. Use these techniques to calculate 3.r? -dxdx erty S where 2 = {(x,y) Osr< 0,0 Sy < 0o). Use proper limit notation when showing your work (sce 7.7 of Calculus Textbook II on Course Ware for review). Bonus: Calculate over the region 1 = {(x,y) 103 x < 0,0 Sy<00). Again, show all work with proper notation.

Answer 1

3x2 Rewrite the integral dydx with proper limits e IG 3x2 -dydx = ESS 3x? -dydx = lim lim ( 3x? 4+02-20 Joert dydx = lim lim (* 1*3x*e****"dx 4+02 -- JOJO lim 4-02-20 e**a!) te-> |dx = lim lim [ei'-) -e>>] -*-* - &** - &*# +1] = lim [0-4** – 0 +1] =-0+1=1 = lim lim 4+02 + L 1 S Sa 1 Rewrite the integra || dxdy with proper limits x2 + y2 exp 2 dxdy = 50 % dxdy x2 + y2 x2 + y2 exp exp 2 2 Change the integral in polar coordinates to get 1 dxdy = L rde dr = lim x2 + y2 exp 2 1 LG ! rde dr 2- JOJO 2 exp exp 2 dr MT lim - -exp 2 2- - im (recople) (+11-1 MT 11 lim -exp 11