009B Sample Final 3, Problem 4 - Revision history

Kayla Murray at 00:43, 3 December 2017

2017-12-03T00:43:55Z

Kayla Murray at 21:47, 12 March 2017

2017-03-12T21:47:12Z

Kayla Murray at 19:29, 3 March 2017

2017-03-03T19:29:38Z

Kayla Murray at 19:23, 3 March 2017

2017-03-03T19:23:20Z

Kayla Murray at 17:18, 3 March 2017

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Kayla Murray at 00:59, 1 March 2017

2017-03-01T00:59:50Z

Kayla Murray at 20:34, 28 February 2017

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Kayla Murray at 02:18, 19 February 2017

2017-02-19T02:18:11Z

Kayla Murray at 17:57, 18 February 2017

2017-02-18T17:57:26Z

Kayla Murray: Created page with "Consider the region bounded by the following two functions: :::::::: $y=2(-x^2+9)$ and
2017-02-18T02:45:16Z

Created page with "Consider the region bounded by the following two functions: :::::::: <math style="vertical-align: -5px">y=2(-x^2+9)</math> and <math styl..."

New page
Consider the region bounded by the following two functions:
:::::::: <math style="vertical-align: -5px">y=2(-x^2+9)</math> and <math style="vertical-align: -4px">y=0</math>.

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.

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'''Solution:'''

'''(a)'''

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'''(b)'''

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'''(c)'''

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[[009B_Sample_Final_3|'''Return to Sample Exam''']]

@@ Line 1: / Line 1: @@
 <span class="exam"> Find the volume of the solid obtained by rotating about the &nbsp;<math>x</math>-axis the region bounded by &nbsp;<math style="vertical-align: -4px">y=\sqrt{1-x^2}</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=0.</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009B Sample Final 3, Problem 4 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009B Sample Final 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 39: / Line 39: @@
 !Step 2: &nbsp;
 |-
-|Using the disk method, the radius of each disk is given by &nbsp;<math style="vertical-align: -3px">r=\sqrt{1-x^2}.</math>
+|Using the disk method, the radius of each disk is given by
 |-
 |Therefore, the volume of the solid is

@@ Line 1: / Line 1: @@
-<span class="exam"> Find the volume of the solid obtained by rotating about the &nbsp;<math>x</math>-axis the region bounded by &nbsp;<math style="vertical-align: -4px">y=\sqrt{1-x^2}</math>&nbsp; and &nbsp;<math>y=0.</math>
+<span class="exam"> Find the volume of the solid obtained by rotating about the &nbsp;<math>x</math>-axis the region bounded by &nbsp;<math style="vertical-align: -4px">y=\sqrt{1-x^2}</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=0.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 21: / Line 21: @@
 !Step 1: &nbsp;
 |-
-|We start by finding the intersection points of the functions <math>y=\sqrt{1-x^2}</math> and <math>y=0.</math>
+|We start by finding the intersection points of the functions &nbsp;<math style="vertical-align: -4px">y=\sqrt{1-x^2}</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">y=0.</math>
 |-
-|We need to solve <math>0=\sqrt{1-x^2}.</math>
+|We need to solve
 |-
 |If we square both sides, we get
 |-
-|<math>0=1-x^2.</math>
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>0=1-x^2.</math>
 |-
-|The solutions to this equation are <math>x=-1</math> and <math>x=1.</math>
+|The solutions to this equation are &nbsp;<math style="vertical-align: -1px">x=-1</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">x=1.</math>
 |-
-|Hence, we are interested in the region between <math>x=-1</math> and <math>x=1.</math>
+|Hence, we are interested in the region between &nbsp;<math style="vertical-align: -1px">x=-1</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">x=1.</math>
 |}
@@ Line 37: / Line 39: @@
 !Step 2: &nbsp;
 |-
-|Using the disk method, the radius of each disk is given by <math>r=\sqrt{1-x^2}.</math>
+|Using the disk method, the radius of each disk is given by &nbsp;<math style="vertical-align: -3px">r=\sqrt{1-x^2}.</math>
 |-
 |Therefore, the volume of the solid is

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+|
 &nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -13px">\int \pi r^2~dx,</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">r</math>&nbsp; is the radius of the disk.
 |}
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 !Step 1: &nbsp;
 |-
+|We start by finding the intersection points of the functions <math>y=\sqrt{1-x^2}</math> and <math>y=0.</math>
 |-
+|If we square both sides, we get
 |-
+|<math>0=1-x^2.</math>
 |-
+|The solutions to this equation are <math>x=-1</math> and <math>x=1.</math>
 |}
@@ Line 39: / Line 37: @@
 !Step 2: &nbsp;
 |-
+|Using the disk method, the radius of each disk is given by <math>r=\sqrt{1-x^2}.</math>
 |-
+|Therefore, the volume of the solid is
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 |}
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 !Final Answer: &nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{4\pi}{3}</math>
 |}
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

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 '''Solution:'''
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-!Step 1: &nbsp;
+!Final Answer: &nbsp;
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+|
 |}
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 20:34, 28 February 2017
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−	<span class="exam"> Find the volume of the solid obtained by rotating about the <math>x</math>-axis the region bounded by <math>y=\sqrt{1-x^2}</math> and <math>y=0.</math>	+	<span class="exam"> Find the volume of the solid obtained by rotating about the  <math>x</math>-axis the region bounded by  <math style="vertical-align: -4px">y=\sqrt{1-x^2}</math>  and  <math>y=0.</math>

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"

← Older revision		Revision as of 17:57, 18 February 2017
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−	<span class="exam">~~Consider~~ the ~~region bounded~~ by the ~~following two functions:~~	+	<span class="exam"> Find the volume of the solid obtained by rotating about the <math>x</math>-axis the region bounded by <math>y=\sqrt{1-x^2}</math> and <math>y=0.</math>
−	~~::::::::~~<~~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">a) Using the lower sum with three rectangles having equal width, approximate the area.~~
−
−	~~<span class="exam">b) Using the upper sum with three rectangles having equal width, approximate the area.~~
−
−	~~<span class="exam">c) Find the actual area of the region.~~

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"