Polar Coordinates and volume?

NetherCraft 0

Use polar coordinates to find the volume of the given solid.

Bounded by the paraboloid z = 7 + 2×2 + 2y2 and the plane z = 13 in the first octant.

2 Answers

  • Using polar coordinates,

    z = 7 + 2x^2 + 2y^2 = 7 + 2r^2.

    This intersects the plane when 7 + 2r^2 = 13

    ==> r^2 = 3

    ==> r = √3, a circle.

    (We take θ in [0, π/2], since we want the region in the first octant, where in particular x, y > 0.)

    So, the volume equals

    ∫∫ (7 + 2x^2 + 2y^2) dA

    = ∫(θ = 0 to π/2) ∫(r = 0 to √3) (7 + 2r^2) * (r dr dθ), converting to polar coordinates

    = ∫(θ = 0 to π/2) dθ * ∫(r = 0 to √3) (7r + 2r^3) dr

    = (π/2) * (7r^2/2 + r^4/2) {for r = 0 to √3}

    = 15π/2.

    I hope this helps!

  • kb, I am following your work, except..

    when we get to this sage…

    ∫∫ (7 + 2x^2 + 2y^2) dA

    shoud it not be?….

    ∫∫ 13 – (7 + 2x^2 + 2y^2) dA

    and converting to cylindrical coordinates.

    ∫∫ (6 – 2r^2)r dr dθ

    = (π/2) * (6r^2/2 – r^4/2) {for r = 0 to √3}

    = 9 pi / 4

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