Question

9. Set up the necessary intervals to find the average value of the function /(x,y)=x' +4y2 - 4 over the ellipse 4x² + y2 = 4. Use technology to evaluate the integral. Give the exact answer.

Answer 1

2.9] Set up necessary intervals to find the curage value of the function flu,y)= x + 4y²-4 over the ellipse 4 x² + y² = 4. Any: Average value of function flu,y) over the Regian D is gilum by fang 1 f(ny) da A(D) D Where Alo) is area of the Region D. • our Region D is bounded by ellipse 40² + y ² - 4 H² + y 2 2. 1 4. (0,2) y2 1 2. N (1) 2 2) . 111,0) Х cef We know that area .. ellipse 12 a2 + y2 ل | = 2 b - 7 ab hence our AlD) = m (1) (2) = 271 Scanned with CamScanner

Now we Evaluate - fang ² + 4y² - 4 ) d A 2T We are going to use Palur Coordinate = e coso y = 2 r Simo And d A = 2 r drdo Im ellipse n² +42 92 12 and narcose, y=brsimo dA=rab dr do ocoL2 T ocre 1 21 -) faug 1 - les cos'o + 43sino - 4)zadado 2T 27 1 1 cos²o + 8 r² sin²o- 8r) dedo 2 T O 0 2 T 244 cosef 8 rt Sim²o - 84² do 4 4 2T 4 CSScanned with CamScanner

RT U 1 Cos’o + 2 Sin?o – 27 do 2T 2 2T 1 2 T ST +2 2 RTV f 1 14 C0520) 1-095190) - 2 ] do z do 22 coszo] do Coszot 1 - Cos20-2 2T 4. 4 4 3 2 T 4 2 T 3 1 + cos2o o] do 2T 4 ♡ 271 - 13) TO 6 + Sinzo 2 T 2 1 X 3 4 2 T to oro :) como 2T 3. U | fang. Ary. 4 Scanned with CamSeanner