Find the average value of f(x, y,z) = xyz over x2 + y2 + z2 2, z > 0, x > 0. y>0

Question

Find the average value of f(x, y,z) = xyz over x2 + y2 + z2 2, z > 0, x > 0. y>0. Evaluate the surface integral (x,y,1) dS, where S is the portion of the paraboloid z = 1 - x2 - y2 that lies over the unit disk. Evaluate the line integral of F(x,y,z) = (x,-y2,z), where c(r) - (sin t, cos t.2t) from t = 0 to t = pi. Express as an equivalent integral in the order dydzdx: Use the transformation u - x + y, v - x - y to find where D is the region enclosed by the lines x+y=0, x+y=l, x-y= 1, x-y=4. Find the surface integral of V x where t(x,y,z) - (z - y, z + x, - x - y) over the portion of the paraboloid z - 9- x2 -y2 above the xy-plane.

Explanation / Answer

Ans 4) 0<x<2, 0<z<4-(x^2), z<y<8-z
Ans 5) 0<U<1, 1<V<4, use jacobian transformation

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