Question

4. (20 pts) Let R be a ring of radius 2 centered at (1,1) and pt, y) = x2 + y2 – 2.– 2y + 2 be a density function. (a) (2 pts) Which equation calculates the total mass? i pardy ii. pds ii. S xpdr + [ updy iv. Izpds + / upas (b) (4 pts) Amy intended to find the total mass. She noticed that the polar coordinates might be suitable, so she set x = 2 cost, y = 2 sint and substitute p as p2 - 2r cost - 2r sint +2. Then she formulated the integral as M = [ (6 - 4 cost – 4 sint). 2 dt. (Hint: What is the para- However she made a mistake. Pleas help her find the mistake. metric curve C in her formulation?)

Answer 1

( The total mass is sp ds = {(x-18? (y-2) de where c is the circle (x-1)? 14-1²-2² (b) She has taken the incorrect procomotrization of the curve, here ering is not centered at 10,0). corect Conect baon paromerization C) is (1+20st, 1+2 sint)=r(t). 2T - M= spas - SP Cact) Ir(t)) at M = 2T T 4x2 dt = 16T (d) Moment along x axis Mx Mx = J (1+2 Bint) por (t)) Ir (oldt = 15 (1+2 sint)dt = 167 [using & sintet =o) My = S xpds - 18 (1+2Gs) at = 16 7 Cost dt = 16 T ce center of mass = (My , Mx ) = (1,1).