Question

y? Evaluate SJ-ye dx dy, over the first quadrant of the ellipse 62 x? 2. + y? 52 =1.

Answer 1

(1) IE W/2 you! + 0 goo + - que Question Let the given integral be Essay loha dady Also the equation of the echlitse is ye (2) Solution: From (1) and (2), we have I=Say we dady - I-Day dady S: (1/2=1 Now we evaluate the integesel given by equation (8) over the first quadeant of the ellipse g? у +3), for each HEN + (4) q? B 91 Blogbil Q A! (-4,0) 00to) Alaio) B! (0,-6) colecally the shaded Legion OABO in the above figure is the best of the ellipse

(Q) ye + OR पुर - har en tu = 1, which lies in the first quadzant of the ellipse =1 Now consider a steiß PQRS, parallel to a- ualis in the shaded legion of the fighe on page - | The deft end of the steip PQRS dies on y-991-3, 7-6, 930 and the light end of the strip PQ RS dies on the ellipse वर zl El- पर reo, Qe= ((7- ) , tes= .. the limits of ale from 0170 or all alleen een Also the StRib PQRS, barallel to 0~9963 in the shaded togion of the figule on page-1 can move from yzo (3–9483) to y=b, therefore the finits of I are from 9ao to yobo ge 1800, que que 69) 4= 011-ben a to a=

(3) s from equation (o), we have I- so gall-wall ay do dy G=0 030 ego gall fod ya yadady ( Nobel y=0 a=0 را all-google dy all-out integrating frest with respect to of, We have - S'y { 30 31 y{ gps =* ${4°41-4) -0% diy any (1) dy Home So (-) dy so dy y=0 2 11 y=0

6 14) y4 462 a [8 - 3-90-03] role the l-4) 252] 11 = < ale х b 986 8 GB .: We have I 8 98% g os s say the hot so we dady over the fiest quadrant of the ellipse yap + 31. aa