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Thread: Volume and Area of a spherical cap

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    Senior Member Vinod's Avatar
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    Volume and Area of a spherical cap

    Hello Forumites,
    I know the formulas for volume and area of a spherical cap of radius r and height h. $V=\frac13\pi*h^2(3*r-h)$ and $A=2*\pi*r*h$. But I don't know how it is derived using multiple integral technique. If any forumite know it he may reply.
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    Re: Volume and Area of a spherical cap

    Aren't you even going to try this yourself? Set up a coordinate system so that the center of the sphere is at (0, 0, 0). The equation of the sphere is $\displaystyle x^2+ y^2+ z^2= r^2$. Take the "cap" to be at the "top" (along the z-axis) so that the top is at (0, 0, r) and the base of the cap is at r- h. Slice that cap with planes parallel to the xy-plane. Each slice is a disk with center at (0, 0, z) and equation $\displaystyle x^2+ y^2= r^2- z^2$ so it has radius $\displaystyle \sqrt{r^2- z^2}$ and area $\displaystyle \pi(r^2- z^2)$. Taking each disk to have thickness "dz" The volume of each disk is $\displaystyle \pi(r^2- z^2)dz$ and the total area is $\displaystyle \pi\int_{r- h}^r r^2- z^2 dz$.
    Thanks from topsquark and Vinod
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    Senior Member Vinod's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by HallsofIvy View Post
    Aren't you even going to try this yourself? Set up a coordinate system so that the center of the sphere is at (0, 0, 0). The equation of the sphere is $\displaystyle x^2+ y^2+ z^2= r^2$. Take the "cap" to be at the "top" (along the z-axis) so that the top is at (0, 0, r) and the base of the cap is at r- h. Slice that cap with planes parallel to the xy-plane. Each slice is a disk with center at (0, 0, z) and equation $\displaystyle x^2+ y^2= r^2- z^2$ so it has radius $\displaystyle \sqrt{r^2- z^2}$ and area $\displaystyle \pi(r^2- z^2)$. Taking each disk to have thickness "dz" The volume of each disk is $\displaystyle \pi(r^2- z^2)dz$ and the total area is $\displaystyle \pi\int_{r- h}^r r^2- z^2 dz$.
    Hello,
    How the total surface area of a spherical cap is calculated? $A=2*π*\sqrt{r^2-z^2}*h$ should be the formula for it's total surface area.
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    Member Walagaster's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Vinod View Post
    Hello,
    How the total surface area of a spherical cap is calculated? $A=2*π*\sqrt{r^2-z^2}*h$ should be the formula for it's total surface area.
    Where did you get that? There should be no $z$ in the formula for surface area. It should only depend on $r$ and $h$.
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    Member Walagaster's Avatar
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    Re: Volume and Area of a spherical cap

    In spherical coordinates, if the radius of the sphere is $\rho = r$, the element of surface area is $dS = r^2\sin\phi~d\phi d\theta$. So for the surface area of the cap try$$
    S = \int_0^{2\pi}\int_0^{\arccos\frac {r - h}r}r^2\sin\phi~d\phi d\theta$$
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    Senior Member Vinod's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Walagaster View Post
    In spherical coordinates, if the radius of the sphere is $\rho = r$, the element of surface area is $dS = r^2\sin\phi~d\phi d\theta$. So for the surface area of the cap try$$
    S = \int_0^{2\pi}\int_0^{\arccos\frac {r - h}r}r^2\sin\phi~d\phi d\theta$$
    Hello,
    Would you show me the double integration which you are talking about?
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    Senior Member Vinod's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Vinod View Post
    Hello,
    Would you show me the double integration which you are talking about?
    I got the following formulas for surface area of spherical cap.

    $S=\int{2\pi y ds }$ Rotation about x-axis $S= \int {2\pi x ds} $ Rotation about y-axis.

    Where $ds=\sqrt{1+(\frac{dy}{dx})^2}dx,$ if y=f(x),$a\leq x\leq b $

    $ds=\sqrt{1+(\frac{dx}{dy})^2}dy,$ if x=h(y), $c\leq y\leq d $
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    Member Walagaster's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Walagaster View Post
    In spherical coordinates, if the radius of the sphere is $\rho = r$, the element of surface area is $dS = r^2\sin\phi~d\phi d\theta$. So for the surface area of the cap try$$
    S = \int_0^{2\pi}\int_0^{\arccos\frac {r - h}r}r^2\sin\phi~d\phi d\theta$$
    Quote Originally Posted by Vinod View Post
    Hello,
    Would you show me the double integration which you are talking about?
    I did show you the double integral. It's right there. Just integrate it.
    Thanks from Vinod
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    Senior Member Vinod's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Walagaster View Post
    I did show you the double integral. It's right there. Just integrate it.
    Hello,
    How did you calculate dS and limits of double integration? Actually I got the 2πrh answer by some other method.
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    Senior Member Vinod's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Walagaster View Post
    I did show you the double integral. It's right there. Just integrate it.
    Hello,
    As far as i know $cos \theta =\frac{r-h}{r}$ and $rsin\phi= (r-h)$.How did you compute dS?
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    MHF Contributor MarkFL's Avatar
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    Re: Volume and Area of a spherical cap

    I would simply set up a solid and surface area of revolution.

    Volume:

    $\displaystyle V=\pi\int_{r-h}^{r} r^2-x^2\,dx=\frac{\pi}{3}h^2(3r-h)$

    Lateral surface area:

    $\displaystyle S=2\pi\int_{r-h}^{r} r\,dx=2\pi hr$
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    Member Walagaster's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by MarkFL View Post
    I would simply set up a solid and surface area of revolution.



    Lateral surface area:

    $\displaystyle S=2\pi\int_{r-h}^{r} r\,dx=2\pi hr$
    I'm thinking that is pretty terse if you are essentially agreeing with the OP that his first formula for $ds$ in post #7 is correct. But it isn't clear to me that Vinod knew how to push that formula through for his problem. He didn't show any work and, although it isn't difficult, you haven't shown him how $y~ds = r~dx$ for this particular problem.
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    Member Walagaster's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Vinod View Post
    Hello,
    As far as i know $cos \theta =\frac{r-h}{r}$ and $rsin\phi= (r-h)$.How did you compute dS?
    I assume you know the volume element in spherical coordinates $dV = \rho^2\sin\phi~d\rho d\phi d\theta$. Just leave off the $d\rho$ to get $dS$ for a sphere of constant radius. The cross section in the picture below shows the range of the $\phi$ variable. Have you worked out the integration using this formula yet?
    Attached Thumbnails Attached Thumbnails Volume and Area of a spherical cap-top.jpg  
    Last edited by Walagaster; Sep 9th 2018 at 02:20 PM.
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    MHF Contributor MarkFL's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Walagaster View Post
    I'm thinking that is pretty terse if you are essentially agreeing with the OP that his first formula for $ds$ in post #7 is correct. But it isn't clear to me that Vinod knew how to push that formula through for his problem. He didn't show any work and, although it isn't difficult, you haven't shown him how $y~ds = r~dx$ for this particular problem.
    I did assume he could make the necessary algebraic simplification(s).
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    Senior Member Vinod's Avatar
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    Re: Volume and Area of a spherical cap

    Quote Originally Posted by Walagaster View Post
    I assume you know the volume element in spherical coordinates $dV = \rho^2\sin\phi~d\rho d\phi d\theta$. Just leave off the $d\rho$ to get $dS$ for a sphere of constant radius. The cross section in the picture below shows the range of the $\phi$ variable. Have you worked out the integration using this formula yet?
    Hello,
    The equation of the right semicircle $x=\sqrt{r^2-y^2},\frac{dx}{dy}=\frac{-y}{\sqrt{r^2-y^2}}=\frac{-y}{x}$

    $ds=\sqrt{1+(\frac{dx}{dy})^2}dy =\sqrt{1+(\frac{-y}{x})^2}dy,= \frac{\sqrt{x^2+y^2}}{x}dy, =\frac{r}{x}dy$

    Then $S_y=2\pi \int_c^d xds = 2\pi \int_{r-h}^{r} xds= 2\pi \int_{r-h}^{r}x*\frac{r}{x}dy=2\pi r|y|_{r-h}^{r}$

    $S_y=2\pi r[r-(r-h)]=2\pi rh$
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