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

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<span class="exam">Consider the region bounded by the following two functions:
 
<span class="exam">Consider the region bounded by the following two functions:
−
::<span class="exam"> <math style="vertical-align: -5px">y=2(-x^2+9)</math> and <math style="vertical-align: -4px">y=0</math>.
+
::<span class="exam"> <math style="vertical-align: -5px">y=2(-x^2+9)</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=0</math>.
  
 
<span class="exam">(a) Using the lower sum with three rectangles having equal width, approximate the area.
 
<span class="exam">(a) Using the lower sum with three rectangles having equal width, approximate the area.
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|Recall:
 
|Recall:
 
|-
 
|-
−
|'''1.''' The height of each rectangle in the lower Riemann sum is given by choosing the minimum <math style="vertical-align: -5px">y</math> value of the left and right endpoints of the rectangle.
+
|'''1.''' The height of each rectangle in the lower Riemann sum is given by choosing  
 
|-
 
|-
−
|'''2.''' The height of each rectangle in the upper Riemann sum is given by choosing the maximum <math style="vertical-align: -5px">y</math> value of the left and right endpoints of the rectangle.
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|&nbsp; &nbsp; &nbsp; &nbsp; the minimum &nbsp;<math style="vertical-align: -5px">y</math>&nbsp; value of the left and right endpoints of the rectangle.
 
|-
 
|-
−
|'''3.''' The area of the region is given by <math style="vertical-align: -14px">\int_a^b y~dx</math> for appropriate values <math style="vertical-align: -4px">a,b</math>.
+
|'''2.''' The height of each rectangle in the upper Riemann sum is given by choosing
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; the maximum &nbsp;<math style="vertical-align: -5px">y</math>&nbsp; value of the left and right endpoints of the rectangle.
 +
|-
 +
|'''3.''' The area of the region is given by  
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -14px">\int_a^b y~dx</math>  
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; for appropriate values &nbsp;<math style="vertical-align: -4px">a,b</math>.
 
|}
 
|}
  
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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 style="vertical-align: -5px">2(-x^2+9)=0</math>. Solving for <math style="vertical-align: 0px">x</math>, we get <math style="vertical-align: 0px">x=\pm 3</math>.
+
|So, we let &nbsp;<math style="vertical-align: -5px">2(-x^2+9)=0</math>.&nbsp; Solving for &nbsp;<math style="vertical-align: 0px">x,</math>&nbsp; we get &nbsp;<math style="vertical-align: 0px">x=\pm 3</math>.
 
|-
 
|-
−
|This means that we need to calculate the Riemann sums over the interval <math style="vertical-align: -5px">[-3,3]</math>.
+
|This means that we need to calculate the Riemann sums over the interval &nbsp;<math style="vertical-align: -5px">[-3,3]</math>.
 
|-
 
|-
 
|
 
|
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|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,
+
|Since the length of our interval is &nbsp;<math style="vertical-align: 0px">6</math>&nbsp; and we are using &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; rectangles,
 
|-
 
|-
−
|each rectangle will have width <math style="vertical-align: 0px">2</math>&thinsp;.
+
|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math>  
 
|-
 
|-
 
|Thus, the lower Riemann sum is
 
|Thus, the lower Riemann sum is
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|As in Part (a), the length of our inteval is <math style="vertical-align: 0px">6</math> and  
+
|As in Part (a), the length of our interval is &nbsp;<math style="vertical-align: 0px">6</math>&nbsp; and  
 
|-
 
|-
−
|each rectangle will have width <math style="vertical-align: 0px">2</math>. (See Step 1 and 2 for '''(a)''')
+
|each rectangle will have width &nbsp;<math style="vertical-align: 0px">2.</math> (See Step 1 and 2 for (a))
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)''' &nbsp;<math style="vertical-align: 0px">32</math>
+
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp;<math style="vertical-align: 0px">32</math>
 
|-
 
|-
−
|'''(b)''' &nbsp;<math style="vertical-align: 0px">96</math>
+
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp;<math style="vertical-align: 0px">96</math>
 
|-
 
|-
−
|'''(c)''' &nbsp;<math style="vertical-align: 0px">72</math>
+
|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp;<math style="vertical-align: 0px">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 09:23, 27 February 2017

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:  
Recall:
1. The height of each rectangle in the lower Riemann sum is given by choosing
        the minimum  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 of the left and right endpoints of the rectangle.
2. The height of each rectangle in the upper Riemann sum is given by choosing
        the maximum  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 of the left and right endpoints of the rectangle.
3. The area of the region 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 y~dx}
        for appropriate values  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,b} .


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=\pm 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 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
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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