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Math Help - Use triple integration in Cylindrical coordinates.

  1. #1
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    Use triple integration in Cylindrical coordinates.

    I really need help with the following question, i dont even know where to start:

    Use triple integration in Cylindrical coordinates to find the volume of the region that lies inside both the sphere of radius 2 centered at the origin and the cylinder of radius 1 with the zaxis as its centre.

    Thanks soo much
    Last edited by mr fantastic; February 22nd 2011 at 05:45 PM. Reason: Title.
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  2. #2
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    Where on the \displaystyle z axis is the centre?
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  3. #3
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    Assuming 'centre' meant 'axis', and doing just the top hemisphere, we have z going from 0 (where it 'starts', on the (x,y) plane) up to \sqrt{4 - r^2} (where it hits the hemisphere). And we have r going from 0 at the centre (z axis), up to 1 where it hits the cylinder. And theta is going a full turn. So...

    \displaystyle{V = \int_0^{2\pi} \int_0^1 \int_0^{\sqrt{4 - r^2}} r\ dz\ dr\ d\theta}

    Just in case a picture helps to follow through from the inside out, we can start bottom left here, integrating r with respect to z...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case z, r or theta as indicated), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).


    Which leaves a couple of blanks to fill. Hope this helps.

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