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Math Help - Calculate the volume of a part of a sphere inside a cylinder

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    MHF Contributor arbolis's Avatar
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    Calculate the volume of a part of a sphere inside a cylinder

    Find the volume of the part of the sphere r^2+z^2=a^2 which is inside the cylinder r=a \sin (\theta) where (r,z,\theta) are the cylindrical coordinates.

    My attempt : I'm not able to visualize the sphere nor the cylinder.
    For the sphere, does r^2=x^2+y^2? If so then I can imagine the sphere. (Centered at the origin (0,0,0) and with radius a).
    For the cylinder \sqrt{x^2+y^2}=a \sin (\theta) I don't know how to go further. I also feel it's useless to convert all in Cartesian coordinates but this way it's easier to me to visualize the volumes.
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    Quote Originally Posted by arbolis View Post
    Find the volume of the part of the sphere r^2+z^2=a^2 which is inside the cylinder r=a \sin (\theta) where (r,z,\theta) are the cylindrical coordinates.

    My attempt : I'm not able to visualize the sphere nor the cylinder.
    For the sphere, does r^2=x^2+y^2? If so then I can imagine the sphere. (Centered at the origin (0,0,0) and with radius a).
    For the cylinder \sqrt{x^2+y^2}=a \sin (\theta) I don't know how to go further. I also feel it's useless to convert all in Cartesian coordinates but this way it's easier to me to visualize the volumes.
    r = a \sin \theta \Rightarrow r = a \frac{y}{r} \Rightarrow r^2 = a y \Rightarrow x^2 + y^2 - ay = 0 .

    Now complete the square etc.
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    MHF Contributor arbolis's Avatar
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    So the sphere is centered at the origin and has a radius of a.
    The cylinder is centered in (0,\frac{a}{2}) and has a radius of \frac{a}{4} and is parallel to the z-axis.
    I now can visualize the volume. However I don't know how to set up the triple integrals. Should I use Cartesian coordinates? Must I find the ellipses of intersection between the sphere and the cylinder? How would you proceed?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by arbolis View Post
    Find the volume of the part of the sphere r^2+z^2=a^2 which is inside the cylinder r=a \sin (\theta) where (r,z,\theta) are the cylindrical coordinates.

    My attempt : I'm not able to visualize the sphere nor the cylinder.
    For the sphere, does r^2=x^2+y^2? If so then I can imagine the sphere. (Centered at the origin (0,0,0) and with radius a).
    For the cylinder \sqrt{x^2+y^2}=a \sin (\theta) I don't know how to go further. I also feel it's useless to convert all in Cartesian coordinates but this way it's easier to me to visualize the volumes.
    r^2 + z^2 = a^2 is simply a sphere of radius a centered at the origin.

    following Mr F's lead, the cylinder is x^2 + \left( y - \frac a2 \right)^2 = \left( \frac a2 \right)^2

    in the xy-plane, the trace of this cylinder is a circle with radius a/2 centered at (0, a/2)

    now, you are bounded above and below by the sphere. we can simply find the volume in the first quadrant, and then multiply the answer by 4.

    hence, we integrate the function 1, with the following limits:

    0 \le x \le \sqrt{ay - y^2}, 0 \le y \le a and 0 \le z \le \sqrt{a^2 - x^2 - y^2}

    and don't forget to multiply by 4


    any questions?
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    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Jhevon View Post
    r^2 + z^2 = a^2 is simply a sphere of radius a centered at the origin.

    following Mr F's lead, the cylinder is x^2 + \left( y - \frac a2 \right)^2 = \left( \frac a2 \right)^2

    in the xy-plane, the trace of this cylinder is a circle with radius a/2 centered at (0, a/2)

    now, you are bounded above and below by the sphere. we can simply find the volume in the first quadrant, and then multiply the answer by 4.

    hence, we integrate the function 1, with the following limits:

    0 \le x \le \sqrt{ay - y^2}, 0 \le y \le a and 0 \le z \le \sqrt{a^2 - x^2 - y^2}

    and don't forget to multiply by 4


    any questions?
    That makes perfect sense, thank you very much.
    I'm having difficulties with \int_0^{a} \sqrt{a^2-x^2-y^2} dy but I don't give up.
    And I'm having some troubles to realize that x goes from 0 to \sqrt{ay-y^2} and z from 0 to \sqrt{a^2-x^2-y^2}. But I'll try to derive it. If I have time I'll try to solve the problem using cylindrical coordinates.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by arbolis View Post
    That makes perfect sense, thank you very much.
    I'm having difficulties with \int_0^{a} \sqrt{a^2-x^2-y^2} dy but I don't give up.
    that's not the integral you want to do first. you leave the one with the constant limits for the last. anyway, an integral like that would conventionally be done using trig substitution

    And I'm having some troubles to realize that x goes from 0 to \sqrt{ay-y^2}
    in the equation for the cylinder, solve for x

    and z from 0 to \sqrt{a^2-x^2-y^2}
    solve for z in the equation of the sphere.

    then note that you only care about the first quadrant.
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    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Jhevon View Post
    that's not the integral you want to do first. you leave the one with the constant limits for the last. anyway, an integral like that would conventionally be done using trig substitution
    Yes I know, it was the second integral, the first being the one with dz.

    in the equation for the cylinder, solve for x

    solve for z in the equation of the sphere.
    ok thanks. I think I won't forget this in future.

    then note that you only care about the first quadrant.
    yes I realized thanks to your explanations.

    In cylindrical coordinates, I have a doubt about the second integral (dr) : V=4 \int_0^{\frac{\pi}{2}} \int_0^{a} \int_0^{\sqrt{a^2-r^2}} dzdr d\theta. (Or is it V=4 \int_0^{\frac{\pi}{2}} \int_0^{\text{Another function}} \int_0^{\sqrt{a^2-r^2}} r dzdr d\theta. ?)
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by arbolis View Post
    Yes I know, it was the second integral, the first being the one with dz.

    ok thanks. I think I won't forget this in future.

    yes I realized thanks to your explanations.

    In cylindrical coordinates, I have a doubt about the second integral (dr) : V=4 \int_0^{\frac{\pi}{2}} \int_0^{a} \int_0^{\sqrt{a^2-r^2}} dzdr d\theta. (Or is it V=4 \int_0^{\frac{\pi}{2}} \int_0^{\text{Another function}} \int_0^{\sqrt{a^2-r^2}} r dzdr d\theta. ?)
    it is the latter. how would you find the limits for r?

    it is harder here since the circle in the xy-plane is not centered at the origin
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    In cylindrical coordinates the integral you want needs a \sin \theta in place of the "another function".
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by the_doc View Post
    In cylindrical coordinates the integral you want needs a \sin \theta in place of the "another function".
    i wanted to see if he would have gotten that himself
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    Quote Originally Posted by Jhevon View Post
    i wanted to see if he would have gotten that himself
    LOL. Fair enough --- I'm off to bed.
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    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Jhevon View Post
    it is the latter. how would you find the limits for r?

    it is harder here since the circle in the xy-plane is not centered at the origin
    I was thinking about a \sqrt{ \text{something}^2-(x\text{another thing} )^2}.

    Quote Originally Posted by the_doc View Post
    In cylindrical coordinates the integral you want needs a \sin \theta in place of the "another function".
    Thanks, but I don't understand why. Could you explain why is it \frac{ay}{r}=a\sin \theta.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by arbolis View Post
    I was thinking about a \sqrt{ \text{something}^2-(x\text{another thing} )^2}.


    Thanks, but I don't understand why. Could you explain why is it \frac{ay}{r}=a\sin \theta.
    r = a \sin \theta was given
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    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Jhevon View Post
    r = a \sin \theta was given
    lol, I can't believe I missed that. That's the change of coordinates. Believe it or not, I realized this a few minutes ago while cooking.

    Edit: I'm stuck on the last integral : V=-\frac{4}{3} a^{\frac{3}{2}} \int _0^{\frac{\pi}{2}} \sin ^{\frac{3}{2}} (\theta) d\theta.
    Last edited by arbolis; June 14th 2009 at 04:21 PM.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by arbolis View Post
    lol, I can't believe I missed that. That's the change of coordinates. Believe it or not, I realized this a few minutes ago while cooking.

    Edit: I'm stuck on the last integral : V=-\frac{4}{3} a^{\frac{3}{2}} \int _0^{\frac{\pi}{2}} \sin ^{\frac{3}{2}} (\theta) d\theta.
    how exactly did you end up with that integral? that guy never showed up when i did it
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