Volume of a Sphere - Revision history

MathAdmin at 19:48, 20 October 2017

2017-10-20T19:48:08Z

MathAdmin at 19:47, 20 October 2017

2017-10-20T19:47:50Z

MathAdmin: Created page with "Let's say that we want to find the volume of a sphere of radius $r$ using volumes of revolution. We know that the equation..."

2017-10-20T16:50:20Z

Created page with "Let's say that we want to find the volume of a sphere of radius <math style="vertical-align: 0px">r</math> using volumes of revolution. We know that the equation..."

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Let's say that we want to find the volume of a sphere of radius  <math style="vertical-align: 0px">r</math>  using volumes of revolution.

We know that the equation of a circle of radius  <math style="vertical-align: 0px">r</math>  centered at the origin is

::<math>x^2+y^2=r^2.</math>

The upper half semicircle is given by  <math style="vertical-align: -5px">y=\sqrt{r^2-x^2}.</math>

[[File:VolSphereRect.png|400px]]

Now, we want to rotate the upper half semicircle around the  <math style="vertical-align: 0px">x</math>-axis. This will give us a sphere of radius  <math style="vertical-align: 0px">r.</math>

[[File:SphereVol2.gif|400px]]

We use the washer/disk method to find the volume of the sphere. The volume of the sphere is

        <math>\begin{array}{rcl}
\displaystyle{V} & = & \displaystyle{\int_{-r}^r \pi (\sqrt{r^2-x^2})^2~dx}\\
&&\\
& = & \displaystyle{\int_{-r}^r \pi (r^2-x^2)~dx}\\
&&\\
& = & \displaystyle{\pi \bigg(r^2x-\frac{x^3}{3}\bigg)\bigg|_{-r}^r}\\
&&\\
& = & \displaystyle{\pi\bigg(r^3-\frac{r^3}{3}\bigg)-\pi\bigg(-r^3+\frac{r^3}{3}\bigg)}\\
&&\\
& = & \displaystyle{\frac{4}{3}\pi r^3.}
\end{array}</math>

Hence, the volume of a sphere of radius  <math style="vertical-align: 0px">r</math>  is

::<math>V=\frac{4}{3}\pi r^3.</math>

@@ Line 11: / Line 11: @@
 Now, we want to rotate the upper half semicircle around the &nbsp;<math style="vertical-align: 0px">x</math>-axis. This will give us a sphere of radius &nbsp;<math style="vertical-align: 0px">r.</math>
-[[File:File:VolSphere_Small.gif|400px]]
+[[File:VolSphere_Small.gif|400px]]
 We use the washer/disk method to find the volume of the sphere. The volume of the sphere is

@@ Line 11: / Line 11: @@
 Now, we want to rotate the upper half semicircle around the &nbsp;<math style="vertical-align: 0px">x</math>-axis. This will give us a sphere of radius &nbsp;<math style="vertical-align: 0px">r.</math>
-[[File:SphereVol2.gif|400px]]
+[[File:File:VolSphere_Small.gif|400px]]
 We use the washer/disk method to find the volume of the sphere. The volume of the sphere is

Volume of a Sphere - Revision history

MathAdmin at 19:48, 20 October 2017

MathAdmin at 19:47, 20 October 2017

MathAdmin: Created page with "Let's say that we want to find the volume of a sphere of radius r using volumes of revolution. We know that the equation..."

MathAdmin: Created page with "Let's say that we want to find the volume of a sphere of radius $r$ using volumes of revolution. We know that the equation..."