Question

The average value of a function f(x, y, z) over a solid region E is defined to be fave = V(E) f(x, y, z) dv where V(E) is the volume of E. For instance, if p is a density function, then Pave is the average density of E. Find the average value of the function f(x, y, z) = 5x2z + 5y2z over the region enclosed by the paraboloid z = 9 – x2 - y2 and the plane z = 0.

Answer 1

solution : $(x,y,z) = 5x+2 + 5y?2. Paraboloid ,z = 9-x²-y? plane, z=0. - paraboloid meets the x-y plane when x² + y² = 9. = x² + y² = 3², where 8=3. when a circle of radius 3 meets the x-y plane ve, Paraboloid Centre at origin. * use cylindrical co-ordinates 60, 0, z). Z9-x? Then, paraboloid becomes and f= 5x²z+ 5y²z. = 5(x² + y²) Z = 58²2. f over the region R, then. - If F is the mean of ff Rody = Spsdr.

Obtain SR.du. 271 3 9-82 SR.du = ss Sraz dr.de 0-0 3-0 250 21 3 a-z? = ' S (2).ds.de 0:0 6:0 " } :(2-) ds.de o (15-rber.de ALEXEY') do $ 6m) - 1) de = 0 - Spide = 2001" = 81 [AT] SR.du = 81.

- Sasar = 1 Gr 2) a da se do 0-0 3-0 220 588 +(63") do DO يام او 1 2 (1-12 07] do 5 C812 18***) do * ] [*2)==&+6)" [ "_1225+ .co و او او 10,935 Co" = 10,935 [an] 16 SR. f.dv = 10,935 T 8

0 = F = [10935"] P = 10,935 T K = $ is of the function given the average value