Difference between revisions of "009B Sample Final 3, Problem 4"
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| − | <span class="exam"> Find the volume of the solid obtained by rotating about the <math>x</math>-axis the region bounded by <math style="vertical-align: -4px">y=\sqrt{1-x^2}</math> and <math>y=0.</math> | + | <span class="exam"> Find the volume of the solid obtained by rotating about the <math>x</math>-axis the region bounded by <math style="vertical-align: -4px">y=\sqrt{1-x^2}</math> and <math style="vertical-align: -4px">y=0.</math> |
{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
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!Step 1: | !Step 1: | ||
|- | |- | ||
| − | |We start by finding the intersection points of the functions <math>y=\sqrt{1-x^2}</math> and <math>y=0.</math> | + | |We start by finding the intersection points of the functions <math style="vertical-align: -4px">y=\sqrt{1-x^2}</math> and <math style="vertical-align: -4px">y=0.</math> |
|- | |- | ||
| − | |We need to solve <math>0=\sqrt{1-x^2}.</math> | + | |We need to solve |
| + | |- | ||
| + | | <math>0=\sqrt{1-x^2}.</math> | ||
|- | |- | ||
|If we square both sides, we get | |If we square both sides, we get | ||
|- | |- | ||
| − | |<math>0=1-x^2.</math> | + | | <math>0=1-x^2.</math> |
|- | |- | ||
| − | |The solutions to this equation are <math>x=-1</math> and <math>x=1.</math> | + | |The solutions to this equation are <math style="vertical-align: -1px">x=-1</math> and <math style="vertical-align: -1px">x=1.</math> |
|- | |- | ||
| − | |Hence, we are interested in the region between <math>x=-1</math> and <math>x=1.</math> | + | |Hence, we are interested in the region between <math style="vertical-align: -1px">x=-1</math> and <math style="vertical-align: -1px">x=1.</math> |
|} | |} | ||
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!Step 2: | !Step 2: | ||
|- | |- | ||
| − | |Using the disk method, the radius of each disk is given by <math>r=\sqrt{1-x^2}.</math> | + | |Using the disk method, the radius of each disk is given by <math style="vertical-align: -3px">r=\sqrt{1-x^2}.</math> |
|- | |- | ||
|Therefore, the volume of the solid is | |Therefore, the volume of the solid is | ||
Revision as of 11:29, 3 March 2017
Find the volume of the solid obtained by rotating about the -axis the region bounded by and 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=0.}
| Foundations: |
|---|
| 1. You can find the intersection points of two functions, say 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 f(x),g(x),} |
|
by setting 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 f(x)=g(x)} and solving for 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. The volume of a solid obtained by rotating a region 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 using disk method 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 \int \pi r^2~dx,} where 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 the radius of the disk. |
Solution:
| Step 1: |
|---|
| We start by finding the intersection points of the functions 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{1-x^2}} and 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=0.} |
| We need to solve |
| 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 0=\sqrt{1-x^2}.} |
| If we square both sides, we get |
| 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 0=1-x^2.} |
| The solutions to this equation are 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=-1} and 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=1.} |
| Hence, we are interested in the region between 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=-1} and 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=1.} |
| Step 2: |
|---|
| Using the disk method, the radius of each disk 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 r=\sqrt{1-x^2}.} |
| Therefore, the volume of the solid 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_{-1}^1 \pi (\sqrt{1-x^2})^2~dx}\\ &&\\ & = & \displaystyle{\int_{-1}^1 \pi (1-x^2)~dx}\\ &&\\ & = & \displaystyle{\pi\bigg(x-\frac{x^3}{3}\bigg)\bigg|_{-1}^1}\\ &&\\ & = & \displaystyle{\pi\bigg(1-\frac{1}{3}\bigg)-\pi\bigg(-1+\frac{1}{3}\bigg)}\\ &&\\ & = & \displaystyle{\frac{4\pi}{3}.} \end{array}} |
| Final Answer: |
|---|
| 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 \frac{4\pi}{3}} |