Difference between revisions of "009B Sample Final 2, Problem 3"
Jump to navigation
Jump to search
Kayla Murray (talk | contribs) |
Kayla Murray (talk | contribs) |
||
| Line 9: | Line 9: | ||
by setting <math style="vertical-align: -5px">f(x)=g(x)</math> and solving for <math style="vertical-align: 0px">x.</math> | by setting <math style="vertical-align: -5px">f(x)=g(x)</math> and solving for <math style="vertical-align: 0px">x.</math> | ||
|- | |- | ||
| − | |'''2.''' The volume of a solid obtained by rotating an area around the <math style="vertical-align: | + | |'''2.''' The volume of a solid obtained by rotating an area around the <math style="vertical-align: 0px">x</math>-axis using the washer method is given by |
|- | |- | ||
| | | | ||
| − | <math style="vertical-align: - | + | <math style="vertical-align: -18px">\int \pi(r_{\text{outer}}^2-r_{\text{inner}}^2)~dx,</math> where <math style="vertical-align: -4px">r_{\text{inner}}</math> is the inner radius of the washer and <math style="vertical-align: -4px">r_{\text{outer}}</math> is the outer radius of the washer. |
|} | |} | ||
| Line 21: | Line 21: | ||
!Step 1: | !Step 1: | ||
|- | |- | ||
| − | |First, we need to find the intersection points of <math>y=x</math> and <math>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>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 33: | Line 33: | ||
\end{array}</math> | \end{array}</math> | ||
|- | |- | ||
| − | |Hence, these two curves intersect at <math>x=0</math> and <math>x=1.</math> | + | |Hence, these two curves intersect at <math style="vertical-align: 0px">x=0</math> and <math style="vertical-align: 0px">x=1.</math> |
|- | |- | ||
| − | |So, we are interested in the region between <math>x=0</math> and <math>x=1.</math> | + | |So, we are interested in the region between <math style="vertical-align: 0px">x=0</math> and <math style="vertical-align: -1px">x=1.</math> |
|} | |} | ||
| Line 43: | Line 43: | ||
|We use the washer method to calculate this volume. | |We use the washer method to calculate this volume. | ||
|- | |- | ||
| − | |The outer radius is <math>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>r_{\text{inner}}=2-x.</math> | + | |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 16:21, 4 March 2017
Find the volume of the solid obtained by rotating the region bounded by the curves 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 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: |
|---|
| 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}} |