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Thread: cylindrical coordinates

  1. #1
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    cylindrical coordinates

    Use cylindrical coordinates to evaluate
    $\displaystyle \int\int\int_{E} (x^3 + xy^2) dV$, where $\displaystyle E$ is the solid in the first octant (i.e. $\displaystyle x, y, z \geq 0$ that lies beneath the paraboloid $\displaystyle z = 1 - x^2 - y^2$.
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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
    $\displaystyle \int\int\int_{E} (x^3 + xy^2) dV$, where $\displaystyle E$ is the solid in the first octant (i.e. $\displaystyle x, y, z \geq 0$ that lies beneath the paraboloid $\displaystyle z = 1 - x^2 - y^2$.
    $\displaystyle x = \cos\theta$
    $\displaystyle y = \sin\theta$
    $\displaystyle z = z$
    $\displaystyle r^2=x^2+y^2$
    $\displaystyle dV=r\,dz\,dr\,d\theta$

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

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

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

    I'll let you integrate it.
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