Question

3. Draw the region D and evaluate the double integral using polar coordinates. (a) SI x + y dA, x2 + y2 D= {(x, y)| x2 + y2 < 1, x + y > 1} D (b) ſ sin(x2 + y2)dA, D is in the third quadrant enclosed by m2 + y2 = 71, x2 + y2 = 27, y=x, y= V3x.

Answer 1

SS aty رب dA chtch {(2,4) / 22 + y² = 1, +971 } y Ehru nty=1 in Polar Co-ordinates =rcoso DA = r dr do y = r Sino nity² = 12 نیا a rag and DE {(ro) nty > - ri Sinot sino' n² + y2 e la Vel Tino tlose er & 1 Sino+loso puno os os The D = {(1, 0) I sino tlose ers?, OLOSTA }

A ) anty n²+ y & (Sinot loso) yar do x sino tloso do (sino + loso) con 1 Sinot coso (sinot loso) o do Sinotoso Sinot loso- I do 1/2 - Coso) t(sino) (م) - 1+1-17 14 2- IT 2. I sin (x² + y²) dA 7 Ouadrant D- In third enclosed by x² + y² = 1, x² + y² = 210 y=4, y = fon اما tyessa y=1 D a x² + y² = [25]? 03

To Polas co-ordinates Region Tz x ² + y² L ZIT So are Jata for o y= xo = ln y = 3 ohy iT + II ao a att (iRegion 3 Quadrant 3) 4 ST Lo L ut 3 ។ So, D= {(,0)) daereo St LO E UT} Sin (a tyl) da ag i sin (rh) v dor do ST so ५प/ и ST nole note s sin (8). rde I sint de Put seat ardra da Viso =) t= IT V=Sun =) t = 20 (-101) 2 (- cost) on so, I sin câty ²) dA = - I do =-(0) =-(0) SH - [ T 17 >T 12