Difference between revisions of "Volume of a Sphere"
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| − | Let's say that we want to find the volume of a sphere of radius <math>r</math> using volumes of revolution. | + | Let's say that we want to find the volume of a sphere of radius <math>r</math> using volumes of revolution. |
| − | We know that the equation of a circle of radius <math>r</math> centered at the origin is | + | We know that the equation of a circle of radius <math>r</math> centered at the origin is |
::<math>x^2+y^2=r^2.</math> | ::<math>x^2+y^2=r^2.</math> | ||
| − | The upper half semicircle is given by | + | The upper half semicircle is given by <math>y=\sqrt{r^2-x^2}.</math> |
(insert picture of semicircle) | (insert picture of semicircle) | ||
| − | Now, we want to rotate the upper half semicircle around the <math>x</math>-axis. This will give us a sphere of radius <math>r.</math> | + | Now, we want to rotate the upper half semicircle around the <math>x</math>-axis. This will give us a sphere of radius <math>r.</math> |
(insert pictures) | (insert pictures) | ||
| Line 27: | Line 27: | ||
\end{array}</math> | \end{array}</math> | ||
| − | Hence, the volume of a sphere of radius <math>r</math> is <math>V=\frac{4}{3}\pi r^3.</math> | + | Hence, the volume of a sphere of radius <math>r</math> is |
| + | |||
| + | ::<math>V=\frac{4}{3}\pi r^3.</math> | ||
Revision as of 09:18, 27 August 2017
Let's say that we want to find the volume of a sphere of radius using volumes of revolution.
We know that the equation of a circle of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} centered at the origin is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2=r^2.}
The upper half semicircle is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sqrt{r^2-x^2}.}
(insert picture of semicircle)
Now, we want to rotate the upper half semicircle around the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis. This will give us a sphere of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r.}
(insert pictures)
We use the washer/disk method to find the volume of the sphere. The volume of the sphere is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}
Hence, the volume of a sphere of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=\frac{4}{3}\pi r^3.}