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Math Help - cylindrical coordinates

  1. #1
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    cylindrical coordinates

    Use cylindrical coordinates to evaluate
    \int\int\int_{E} (x^3 + xy^2) dV, where E is the solid in the first octant (i.e. x, y, z \geq 0 that lies beneath the paraboloid z = 1 - x^2 - y^2.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by wik_chick88 View Post
    Use cylindrical coordinates to evaluate
    \int\int\int_{E} (x^3 + xy^2) dV, where E is the solid in the first octant (i.e. x, y, z \geq 0 that lies beneath the paraboloid z = 1 - x^2 - y^2.
    x = \cos\theta
    y = \sin\theta
    z = z
    r^2=x^2+y^2
    dV=r\,dz\,dr\,d\theta

    So the bounds are:
    \theta: 0\to\pi/2
    r: 0\to 1
    z: 0\to 1-r^2

    So, (x^3 + xy^2)\,dV = x(x^2+y^2)\,dV = r^2\cos(\theta)r\,dz\,dr\,d\theta

    Your integral is \int_0^{\pi/2}\int_0^1\int_0^{1-r^2}r^3\cos(\theta)\,dz\,dr\,d\theta

    I'll let you integrate it.
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