009B Sample Final 3, Problem 7 - Revision history

Kayla Murray at 00:49, 3 December 2017

2017-12-03T00:49:52Z

2017-03-18T20:04:03Z

2017-03-02T18:17:17Z

2017-03-02T18:12:45Z

2017-03-01T17:21:04Z

2017-03-01T00:58:48Z

2017-02-19T02:16:55Z

2017-02-18T18:04:16Z

@@ Line 3: / Line 3: @@
 ::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009B Sample Final 3, Problem 7 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009B Sample Final 3, Problem 7 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 65: / Line 65: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp;converges
+|&nbsp; &nbsp; &nbsp; &nbsp;converges (by the Direct Comparison Test for Improper Integrals)
 |}
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 25: / Line 25: @@
 |We use the Direct Comparison Test for Improper Integrals.
 |-
-|For all &nbsp;<math>x</math>&nbsp; in &nbsp;<math>[1,\infty),</math>
+|For all &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; in &nbsp;<math style="vertical-align: -5px">[1,\infty),</math>
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>0\le \frac{\sin^2(x)}{x^3} \le \frac{1}{x^3}.</math>
@@ Line 31: / Line 31: @@
 |Also,
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{\sin^2(x)}{x^3}</math>&nbsp; and &nbsp;<math>\frac{1}{x^3}</math>
+|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -15px">\frac{\sin^2(x)}{x^3}</math>&nbsp; and &nbsp;<math style="vertical-align: -15px">\frac{1}{x^3}</math>
 |-
-|are continuous on &nbsp;<math>[1,\infty).</math>
+|are continuous on &nbsp;<math style="vertical-align: -5px">[1,\infty).</math>
 |}
@@ Line 51: / Line 51: @@
 \end{array}</math>
 |-
-|Since &nbsp;<math>\int_1^\infty \frac{1}{x^3}~dx</math>&nbsp; converges,
+|Since &nbsp;<math style="vertical-align: -15px">\int_1^\infty \frac{1}{x^3}~dx</math>&nbsp; converges,
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp;<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>

@@ Line 23: / Line 23: @@
 !Step 1: &nbsp;
 |-
+|We use the Direct Comparison Test for Improper Integrals.
 |-
+|Also,
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{\sin^2(x)}{x^3}</math>&nbsp; and &nbsp;<math>\frac{1}{x^3}</math>
 |-
+|are continuous on &nbsp;<math>[1,\infty).</math>
 |}
@@ Line 35: / Line 39: @@
 !Step 2: &nbsp;
 |-
+|Now, we have
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
 |-
+|converges by the Direct Comparison Test for Improper Integrals.
 |-
+|
@@ Line 49: / Line 65: @@
 !Final Answer: &nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;converges
 |}
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 17: / Line 17: @@
 '''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 44: / Line 42: @@
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
+!Final Answer: &nbsp;
 |-
+|
 |}
 [[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 6: / Line 6: @@
 !Foundations: &nbsp;
 |-
+|'''Direct Comparison Test for Improper Integrals'''
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; Let &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">g</math>&nbsp; be continuous on &nbsp;<math style="vertical-align: -5px">[a,\infty)</math>
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; where &nbsp;<math style="vertical-align: -5px">0\le f(x)\le g(x)</math>&nbsp; for all &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; in &nbsp;<math style="vertical-align: -5px">[a,\infty).</math>
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;'''1.'''&nbsp; If &nbsp;<math style="vertical-align: -14px">\int_a^\infty g(x)~dx</math>&nbsp; converges, then &nbsp;<math style="vertical-align: -14px">\int_a^\infty f(x)~dx</math>&nbsp; converges.
 |}

@@ Line 1: / Line 1: @@
 <span class="exam">Does the following integral converge or diverge? Prove your answer!
-::::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
+::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 14: / Line 14: @@
+|
 |}
 '''Solution:'''
@@ Line 78: / Line 79: @@
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"