Difference between revisions of "009B Sample Final 2, Problem 3"
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|First, we need to find the intersection points of <math style="vertical-align: -5px">y=x</math> and <math style="vertical-align: -5px">y=x^2.</math> | |First, we need to find the intersection points of <math style="vertical-align: -5px">y=x</math> and <math style="vertical-align: -5px">y=x^2.</math> | ||
|- | |- | ||
| − | |To do this, we need to solve <math style="vertical-align: 0px">x=x^2.</math> | + | |To do this, we need to solve |
| + | |- | ||
| + | | <math style="vertical-align: 0px">x=x^2.</math> | ||
|- | |- | ||
|Moving all the terms on one side of the equation, we get | |Moving all the terms on one side of the equation, we get | ||
| Line 43: | Line 45: | ||
|We use the washer method to calculate this volume. | |We use the washer method to calculate this volume. | ||
|- | |- | ||
| − | |The outer radius is <math style="vertical-align: -4px">r_{\text{outer}}=2-x^2</math> and | + | |The outer radius is |
| + | |- | ||
| + | | <math style="vertical-align: -4px">r_{\text{outer}}=2-x^2</math> | ||
| + | |- | ||
| + | |and the inner radius is | ||
|- | |- | ||
| − | | | + | | <math style="vertical-align: -4px">r_{\text{inner}}=2-x.</math> |
|- | |- | ||
|Therefore, the volume of the solid is | |Therefore, the volume of the solid is | ||
Revision as of 13:29, 12 March 2017
Find the volume of the solid obtained by rotating the region bounded by the curves 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=x} 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=x^2} about the line 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=2.}
| 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 an area 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 the washer 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_{\text{outer}}^2-r_{\text{inner}}^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_{\text{inner}}} is the inner radius of the washer 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 r_{\text{outer}}} is the outer radius of the washer. |
Solution:
| Step 1: |
|---|
| First, we need to find the intersection points of 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=x} 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=x^2.} |
| To do this, 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 x=x^2.} |
| Moving all the terms on one side of the equation, 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 \begin{array}{rcl} \displaystyle{0} & = & \displaystyle{x^2-x}\\ &&\\ & = & \displaystyle{x(x-1).} \end{array}} |
| Hence, these two curves intersect at 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=0} 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.} |
| So, 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=0} and |
| Step 2: |
|---|
| We use the washer method to calculate this volume. |
| The outer radius 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 r_{\text{outer}}=2-x^2} |
| and the inner radius 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 r_{\text{inner}}=2-x.} |
| 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_0^1 \pi(r_{\text{outer}}^2-r_{\text{inner}}^2)~dx}\\ &&\\ & = & \displaystyle{\int_0^1 \pi((2-x^2)^2-(2-x)^2)~dx.} \end{array}} |
| Step 3: |
|---|
| Now, we integrate to 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 \begin{array}{rcl} \displaystyle{V} & = & \displaystyle{\pi \int_0^1 ((4-4x^2+x^4)-(4-4x+x^2))~dx}\\ &&\\ & = & \displaystyle{\pi \int_0^1 (4x-5x^2+x^4)~dx}\\ &&\\ & = & \displaystyle{\pi\bigg(2x^2-\frac{5x^3}{3}+\frac{x^5}{5}\bigg)\bigg|_0^1}\\ &&\\ & = & \displaystyle{\pi\bigg(2-\frac{5}{3}+\frac{1}{5}\bigg)-0}\\ &&\\ & = & \displaystyle{\frac{8\pi}{15}.} \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{8\pi}{15}} |