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

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|We need to set these two equations equal in order to find the intersection points of these functions.
 
|We need to set these two equations equal in order to find the intersection points of these functions.
 
|-
 
|-
−
|So, we let <math>2(-x^2+9)=0</math>. Solving for <math>x</math>, we get <math>x=-3,3</math>.
+
|So, we let <math style="vertical-align: -6px">2(-x^2+9)=0</math>. Solving for <math>x</math>, we get <math>x=-3,3</math>.
 
|-
 
|-
 
|This means that we need to calculate the Riemann sums over the interval <math>[-3,3]</math>.
 
|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;
 
|-
 
|-
−
|Since the length of our interval is <math>6</math> and we are using <math>3</math> rectangles,
+
|Since the length of our interval is <math style="vertical-align: 0px">6</math> and we are using <math style="vertical-align: 0px">3</math> rectangles,
 
|-
 
|-
−
|each rectangle will have width <math>2</math>.  
+
|each rectangle will have width <math style="vertical-align: 0px">2</math>.  
 
|-
 
|-
 
|Thus, the lower Riemann sum is
 
|Thus, the lower Riemann sum is
 
|-
 
|-
−
|<math>2(f(-3)+f(-1)+f(3))=2(0+16+0)=32</math>.
+
|
 +
::<math>2(f(-3)+f(-1)+f(3))=2(0+16+0)=32</math>.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|As in Part (a), the length of our inteval is <math>6</math> and  
+
|As in Part (a), the length of our inteval is <math style="vertical-align: 0px">6</math> and  
 
|-
 
|-
−
|each rectangle will have width <math>2</math>. (See Step 1 and 2 for part (a))
+
|each rectangle will have width <math style="vertical-align: 0px">2</math>. (See Step 1 and 2 for '''(a)''')
 
|}
 
|}
  
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|Thus, the upper Riemann sum is  
 
|Thus, the upper Riemann sum is  
 
|-
 
|-
−
|<math>2(f(-1)+f(-1)+f(1))=2(16+16+16)=96</math>
+
|
 +
::<math>2(f(-1)+f(-1)+f(1))=2(16+16+16)=96</math>
 
|}
 
|}
  
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|To find the actual area of the region, we need to calculate
 
|To find the actual area of the region, we need to calculate
 
|-
 
|-
−
|<math>\int_{-3}^3 2(-x^2+9)~dx</math>
+
|
 +
::<math>\int_{-3}^3 2(-x^2+9)~dx</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)''' <math>64</math>
+
|'''(a)''' <math>32</math>
 
|-
 
|-
−
|'''(b)''' <math>64</math>
+
|'''(b)''' <math>96</math>
 
|-
 
|-
 
|'''(c)''' <math>72</math>
 
|'''(c)''' <math>72</math>
 
|}
 
|}
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 16:08, 22 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 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}

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 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(-x^2+9)=0} . 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} , 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 x=-3,3} .
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(3))=2(0+16+0)=32} .

(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 (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(1))=2(16+16+16)=96}

(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 32}
(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 96}
(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}

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