022 Exam 2 Sample A, Problem 6 - Revision history

MathAdmin at 14:54, 16 May 2015

2015-05-16T14:54:34Z

2015-05-16T03:15:48Z

2015-05-16T03:15:20Z

2015-05-16T03:15:05Z

2015-05-15T23:36:33Z

2015-05-15T18:05:55Z

2015-05-15T17:37:14Z

@@ Line 7: / Line 7: @@
 |-
+|
-::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C.</math>
+::<math style="vertical-align: -70%;">\int x^n\,dx\,=\,\frac{x^{n+1}}{n+1} +C,</math>&thinsp; for <math style="vertical-align: -25%;">n\neq -1.</math>
 |-
 |Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: -4%">x = 1</math> and <math style="vertical-align: -2%">x = 4</math>.
@@ Line 32: / Line 33: @@
 \displaystyle{\int_1^{\,4} \frac{8}{\sqrt{x}}\,dx} & = & \displaystyle {\int_1^{\,4} 8x^{-1/2}\,dx}\\
 \\
-& = & \displaystyle{\frac{8 x^{1/2}}{2} \Bigr|_{x=1}^4}\\
+& = & \displaystyle{\frac{8 x^{1/2}}{1/2} \Bigr|_{x=1}^4}\\
 \\
-& = & 4x^{1/2}  \Bigr|_{x=1}^4.
+& = & 16x^{1/2}  \Bigr|_{x=1}^4.
 \end{array}</math>
 |}
@@ Line 44: / Line 45: @@
 |-
+|
-::<math>4x^{1/2}  \Bigr|_{x=1}^4\,=\,4\cdot 4^{1/2} - 4\cdot 1^{1/2} \,=\, 8 - 4 \,=\, 4.</math>
+::<math>16x^{1/2}  \Bigr|_{x=1}^4\,=\,16\cdot 4^{1/2} - 16\cdot 1^{1/2} \,=\, 32 - 16 \,=\, 16.</math>
 |}
@@ Line 51: / Line 52: @@
 |-
+|
-::<math>\int_1^{\,4} \frac{8}{\sqrt{x}} \,dx\,=\,4.</math>
+::<math>\int_1^{\,4} \frac{8}{\sqrt{x}} \,dx\,=\,16.</math>
 |}
 [[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

@@ Line 9: / Line 9: @@
 ::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C.</math>
 |-
-|Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: 0%">x = 1</math> and <math style="vertical-align: 0%">x = 4</math>.
+|Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: -4%">x = 1</math> and <math style="vertical-align: -2%">x = 4</math>.
 |}

@@ Line 9: / Line 9: @@
 ::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C.</math>
 |-
-|Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: -5%">x = 1</math> and <math style="vertical-align: -5%">x = 4</math>.
+|Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: 0%">x = 1</math> and <math style="vertical-align: 0%">x = 4</math>.
 |}

@@ Line 7: / Line 7: @@
 |-
+|
-::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>
+::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C.</math>
 |-
-|For setup of the problem we need to integrate the region between the x - axis, the curve, x = 1, and x = 4.
+|Geometrically, we need to integrate the region between the <math style="vertical-align: 0%">x</math>-axis, the curve, and the vertical lines <math style="vertical-align: -5%">x = 1</math> and <math style="vertical-align: -5%">x = 4</math>.
 |}

@@ Line 4: / Line 4: @@
 !Foundations: &nbsp;
 |-
-|For solving the problem we only require the use of the power rule for integration:
+|For solving the problem, we only require the use of the power rule for integration:
 |-
 |<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>
 |-
 |For setup of the problem we need to integrate the region between the x - axis, the curve, x = 1, and x = 4.
@@ Line 18: / Line 19: @@
 |Set up the integral:
 |-
 |<math>\int_1^4 \frac{8}{\sqrt{x}} dx</math>
 |}
@@ Line 28: / Line 30: @@
+|
 ::<math>\begin{array}{rcl}
-\int_1^4 \frac{8}{\sqrt{x}}dx & = & \frac{8 x^{1/2}}{2} \vert_1^4\\
+\displaystyle{\int_1^{\,4} \frac{8}{\sqrt{x}}\,dx} & = & \displaystyle {\int_1^{\,4} 8x^{-1/2}\,dx}\\
-& = & 4x^{1/2} \vert_1^4
+\\
 \end{array}</math>
 |}
@@ Line 38: / Line 43: @@
 | Now we need to evaluate to get:
 |-
 | <math>4\cdot 4^{1/2} - 4\cdot 1^{1/2} = 8 - 4 = 4</math>
 |}
@@ Line 44: / Line 50: @@
 !Final Answer: &nbsp;
 |-
 |4
 |}
 [[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]