Difference between revisions of "009B Sample Final 1, Problem 1"

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!Step 1:    
 
!Step 1:    
 
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|We need to set these two equations equal in order to find the intersection points of these functions.
 
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|So, we let <math>2(-x^2+9)=0</math>. Solving for <math>x</math>, we get <math>x=-3,3</math>.
 
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|This means that we need to calculate the Riemann sums over the interval <math>[-3,3]</math>.
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Since the length of our interval is <math>6</math> and we are using <math>3</math> rectangles,
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|each rectangle will have width <math>2</math>.
 
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|Thus, the lower Riemann sum is
 
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|<math>2(f(-3)+f(-1)+f(1))=2(0+16+16)=64</math>.
 
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|As in Part (a), the length of our inteval is <math>6</math> and
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|each rectangle will have width <math>2</math>. (See Step 1 and 2 for part (a))
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Thus, the upper Riemann sum is
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!Step 3: &nbsp;
 
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|<math>2(f(-1)+f(1)+f(3))=2(16+16+0)=64</math>
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|'''(a)''' <math>64</math>
 
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|'''(c)''' <math>72</math>
 
|'''(c)''' <math>72</math>
 
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[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 18:02, 4 February 2016

Consider the region bounded by the following 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 y=2(-x^2+9)} and

a) Using the lower sum with three rectangles having equal width , approximate the area.

b) Using the upper sum with three rectangles having equal width, approximate the area.

c) Find the actual area of the region.

Foundations:  
Link to Riemann sums page

Solution:

(a)

Step 1:  
We need to set these two equations equal in order to find the intersection points of these functions.
So, we let . Solving for , we get .
This means that we need to calculate the Riemann sums over 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 [-3,3]} .
Step 2:  
Since the length of our interval 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 6} and we are using 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 3} rectangles,
each rectangle will have width 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} .
Thus, the lower Riemann sum 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 2(f(-3)+f(-1)+f(1))=2(0+16+16)=64} .

(b)

Step 1:  
As in Part (a), the length of our inteval 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 6} and
each rectangle will have width 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} . (See Step 1 and 2 for part (a))
Step 2:  
Thus, the upper Riemann sum 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 2(f(-1)+f(1)+f(3))=2(16+16+0)=64}

(c)

Step 1:  
To find the actual area of the region, we need to calculate
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_{-3}^3 2(-x^2+9)~dx}
Step 2:  
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{\int_{-3}^3 2(-x^2+9)~dx} & = & \displaystyle{2\bigg(\frac{-x^3}{3}+9x\bigg)\bigg|_{-3}^3}\\ &&\\ & = & \displaystyle{2\bigg(\frac{-3^3}{3}+9\times 3\bigg)-2\bigg(\frac{-(-3)^3}{3}+9(-3)\bigg)}\\ &&\\ & = & \displaystyle{2(-9+27)-2(9-27)}\\ &&\\ & = & \displaystyle{2(18)-2(-18)}\\ &&\\ & = & \displaystyle{72}\\ \end{array}}
Final Answer:  
(a) 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 64}
(b) 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 64}
(c) 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 72}

Return to Sample Exam

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