Difference between revisions of "009B Sample Final 2, Problem 2"

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!Step 1:    
 
!Step 1:    
 
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|-
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|First, we need to find the intersection points of these two curves.
 
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|To do this, we set
 
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>3x-x^2=2x^3-x^2-5x.</math>
 
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|-
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|Getting all the terms on one side of the equation, we get
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{0} & = & \displaystyle{2x^3-8x}\\
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&&\\
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& = & \displaystyle{2x(x^2-4)}\\
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&&\\
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& = & \displaystyle{2x(x-2)(x+2).}
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\end{array}</math>
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|-
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|Therefore, we get that these two curves intersect at &nbsp;<math>x=-2,~x=0,~x=2.</math>
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|-
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|Hence, the region we are interested in occurs between &nbsp;<math>x=-2</math>&nbsp; and &nbsp;<math>x=2.</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Since the curves intersect also intersect at &nbsp;<math>x=0,</math>&nbsp; this breaks our region up into two parts,
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|-
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|which correspond to the interval &nbsp;<math>[-2,0]</math>&nbsp; and &nbsp;<math>[0,2].</math>
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|-
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|Now, in each of the regions we need to determine which curve has the higher <math>y</math> value.
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|-
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|To figure this out, we use test points in each interval.
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|-
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|For <math>x=-1,</math> we have
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math> y=3(-1)-(-1)^2=-4</math>&nbsp; and &nbsp;<math>y=2(-1)^3-(-1)^2-5(-1)=2.</math>
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|-
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|For <math>x=1,</math> we have
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math> y=3(1)-(1)^2=2</math>&nbsp; and &nbsp;<math>y=2(1)^3-(1)^2-5(1)=-4.</math>
 
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|  
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|Hence, the area <math>A</math> of the region bounded by these two curves is given by
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>A=\int_{-2}^0 (2x^3-x^2-5x)-(3x-x^2)~dx+\int_0^2 (3x-x^2)-(2x^3-x^2-5x)~dx.</math>
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 3: &nbsp;
 
|-
 
|-
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|Now, we integrate to get
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{A} & = & \displaystyle{\int_{-2}^0 (2x^3-8x)~dx+\int_0^2 (-2x^3+8x)~dx}\\
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&&\\
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& = & \displaystyle{\bigg(\frac{x^4}{2}-4x^2\bigg)\bigg|_{-2}^0+\bigg(\frac{-x^4}{2}+4x^2\bigg)\bigg|_0^2}\\
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&&\\
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& = & \displaystyle{0-\bigg(\frac{(-2)^4}{2}-4(-2)^2\bigg)+\bigg(\frac{-2^4}{2}+4(2)^2\bigg)-0}\\
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&&\\
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& = & \displaystyle{-(8-16)+(-8+16)}\\
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&&\\
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& = & \displaystyle{16.}
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\end{array}</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>16</math>
 
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[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:57, 4 March 2017

Find the area of the region between the two curves    and  

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 area between two 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 f(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 g(x),}   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_a^b f(x)-g(x)~dx}

       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 a\leq x\leq b,}   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 f(x)}   is the upper function 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 g(x)}   is the lower function.


Solution:

Step 1:  
First, we need to find the intersection points of these two curves.
To do this, we set
       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 3x-x^2=2x^3-x^2-5x.}
Getting 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{2x^3-8x}\\ &&\\ & = & \displaystyle{2x(x^2-4)}\\ &&\\ & = & \displaystyle{2x(x-2)(x+2).} \end{array}}
Therefore, we get that 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=-2,~x=0,~x=2.}
Hence, the region we are interested in occurs 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=-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 x=2.}
Step 2:  
Since the curves intersect also 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,}   this breaks our region up into two parts,
which correspond to the interval  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 [-2,0]}   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 [0,2].}
Now, in each of the regions we need to determine which curve has the higher 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} value.
To figure this out, we use test points in each interval.
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=-1,} we have
        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=3(-1)-(-1)^2=-4}   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=2(-1)^3-(-1)^2-5(-1)=2.}
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=1,} we have
        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=3(1)-(1)^2=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=2(1)^3-(1)^2-5(1)=-4.}
Hence, the area 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 A} of the region bounded by these two curves 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 A=\int_{-2}^0 (2x^3-x^2-5x)-(3x-x^2)~dx+\int_0^2 (3x-x^2)-(2x^3-x^2-5x)~dx.}
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{A} & = & \displaystyle{\int_{-2}^0 (2x^3-8x)~dx+\int_0^2 (-2x^3+8x)~dx}\\ &&\\ & = & \displaystyle{\bigg(\frac{x^4}{2}-4x^2\bigg)\bigg|_{-2}^0+\bigg(\frac{-x^4}{2}+4x^2\bigg)\bigg|_0^2}\\ &&\\ & = & \displaystyle{0-\bigg(\frac{(-2)^4}{2}-4(-2)^2\bigg)+\bigg(\frac{-2^4}{2}+4(2)^2\bigg)-0}\\ &&\\ & = & \displaystyle{-(8-16)+(-8+16)}\\ &&\\ & = & \displaystyle{16.} \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 16}

Return to Sample Exam

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