022 Exam 2 Sample B, Problem 6 - Revision history

MathAdmin at 00:29, 18 May 2015

2015-05-18T00:29:51Z

MathAdmin at 00:29, 18 May 2015

2015-05-18T00:29:36Z

MathAdmin at 00:29, 18 May 2015

2015-05-18T00:29:21Z

MathAdmin at 00:26, 18 May 2015

2015-05-18T00:26:11Z

MathAdmin at 14:58, 17 May 2015

2015-05-17T14:58:33Z

MathAdmin at 14:51, 17 May 2015

2015-05-17T14:51:24Z

MathAdmin at 01:08, 16 May 2015

2015-05-16T01:08:55Z

MathAdmin: Created page with "Find the area under the curve of $y = 6x^2 + 2x$ between the $y$ -axi..."

2015-05-16T01:04:14Z

Created page with "<span class="exam">Find the area under the curve of <math style="vertical-align: -13%">y = 6x^2 + 2x</math> between the <math style="vertical-align: -15%">y</math>-axi..."

New page

<span class="exam">Find the area under the curve of  <math style="vertical-align: -13%">y = 6x^2 + 2x</math> between the <math style="vertical-align: -15%">y</math>-axis and <math style="vertical-align: -1%">x = 2</math>.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|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 = 0 (the y-axis), and x = 4.
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|Set up the integral:
|-
|
::<math>\int_0^{\,2} 6x^2 + 2x \,dx.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|Using the power rule we have:
|-
|
::<math>\begin{array}{rcl}
\int _0^2 6x^2+2x \,dx &=& 2x^3+x^2 \Bigr|_0^2 \\
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
|-
| Now we need to evaluate to get:
|-
|
::<math>2x^3 + x^2 \Bigr|_0^2 = (2(2)^3+(2)^2)-(0+0) = 20.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|
::<math>\int_0^{\,2} 6x^2 + 2x \,dx\,=\,20.</math>
|}

[[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]

@@ Line 7: / Line 7: @@
 |-
+|
-::<math style="vertical-align: -80%;">\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>&thinsp; for <math style="vertical-align: -23%;">n\neq -1</math>,
+::<math style="vertical-align: -70%;">\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>&thinsp; for <math style="vertical-align: -23%;">n\neq -1</math>,
 |-
 |For setup of the problem we need to integrate the region between the x - axis, the curve, <math style="vertical-align: 0%">x = 0</math> (the y-axis), and <math style="vertical-align: 0%">x = 2</math>.

@@ Line 7: / Line 7: @@
 |-
+|
-::<math style="vertical-align: -90%;">\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>&thinsp; for <math style="vertical-align: -27%;">n\neq -1</math>,
+::<math style="vertical-align: -80%;">\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>&thinsp; for <math style="vertical-align: -23%;">n\neq -1</math>,
 |-
 |For setup of the problem we need to integrate the region between the x - axis, the curve, <math style="vertical-align: 0%">x = 0</math> (the y-axis), and <math style="vertical-align: 0%">x = 2</math>.

@@ Line 7: / Line 7: @@
 |-
+|
-::<math>\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>
+::<math style="vertical-align: -90%;">\int x^n dn = \frac{x^{n+1}}{n+1} + C</math>&thinsp; for <math style="vertical-align: -27%;">n\neq -1</math>,
 |-
 |For setup of the problem we need to integrate the region between the x - axis, the curve, <math style="vertical-align: 0%">x = 0</math> (the y-axis), and <math style="vertical-align: 0%">x = 2</math>.

@@ Line 20: / Line 20: @@
 |-
+|
-::<math>\int_0^{\,2} 6x^2 + 2x \,dx.</math>
+::<math>\int_0^{2} 6x^2 + 2x \,dx.</math>
 |}
@@ Line 35: / Line 35: @@
 !Step 3: &nbsp;
 |-
-|FInally, we need to evaluate:
+|Finally, we need to evaluate:
 |-
+|

@@ Line 9: / Line 9: @@
 ::<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 = 0 (the y-axis), and x = 2.
+|For setup of the problem we need to integrate the region between the x - axis, the curve, <math style="vertical-align: 0%">x = 0</math> (the y-axis), and <math style="vertical-align: 0%">x = 2</math>.
 |}
@@ Line 29: / Line 29: @@
 |-
+|
-::<math>\begin{array}{rcl}
+::<math>\int _0^2 6x^2+2x \,dx \,=\, 6\cdot \frac{x^3}{3}+2\cdot \frac{x^2}{2} \Bigr|_{x\,=\,0}^2\,=\,2x^3+x^2 \Bigr|_{x\,=\,0}^2. </math>
 |}
@@ Line 37: / Line 35: @@
 !Step 3: &nbsp;
 |-
-| Now we need to evaluate to get:
+|FInally, we need to evaluate:
 |-
+|
-::<math>2x^3 + x^2 \Bigr|_0^2 = (2(2)^3+(2)^2)-(0+0) = 20.</math>
+::<math>2x^3 + x^2 \Bigr|_{x\,=\,0}^2 = (2(2)^3+(2)^2)-(0+0) = 20.</math>
 |}
@@ Line 46: / Line 44: @@
 !Final Answer: &nbsp;
 |-
 |<math>20</math>
 |}
 [[022_Exam_2_Sample_B|'''<u>Return to Sample Exam</u>''']]

022 Exam 2 Sample B, Problem 6 - Revision history

MathAdmin at 00:29, 18 May 2015

MathAdmin at 00:29, 18 May 2015

MathAdmin at 00:29, 18 May 2015

MathAdmin at 00:26, 18 May 2015

MathAdmin at 14:58, 17 May 2015

MathAdmin at 14:51, 17 May 2015

MathAdmin at 01:08, 16 May 2015

MathAdmin: Created page with "Find the area under the curve of y = 6x^2 + 2x between the y-axi..."

MathAdmin: Created page with "Find the area under the curve of $y = 6x^2 + 2x$ between the $y$ -axi..."