Rate of flow problem?

A fluid has density 600 km/m3 and flows with velocity bar(v) = x bar(i) + y bar(j) + z bar(k), where x, y, and z are measured in meters, and the components of bar(v) are measured in meters per second. Find the rate of flow outward through the part of the paraboloid z = 36 - x^2 - y^2 that lies above the xy plane.

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  • kb
    Lv 7
    9 years ago
    Favorite Answer

    We want to compute ∫∫ 600<x,y,z> · dS

    = ∫∫ 600<x,y,z> · <-z_x, -z_y, 1> dA, using cartesian coordinates

    = ∫∫ 600<x, y, 36 - x^2 - y^2> · <2x, 2y, 1> dA

    = ∫∫ 600(x^2 + y^2 + 36) dA.

    Since the region of integration is the interior of x^2 + y^2 = 36, convert to polar coordinates:

    ∫(θ = 0 to 2π) ∫(r = 0 to 6) 600(r^2 + 36) * r dr dθ

    = 2π ∫(r = 0 to 6) 600(r^3 + 36r) dr

    = 2π 600(r^4/4 + 18r^2) {for r = 0 to 6}

    = 1166400π.

    I hope this helps!

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