Difference between revisions of "022 Exam 2 Sample B, Problem 6"

Latest revision as of 16:29, 17 May 2015

Find the area under the curve of $y=6x^{2}+2x$ between the $y$ -axis and $x=2$ .

Foundations:
For solving the problem, we only require the use of the power rule for integration:
$\int x^{n}dn={\frac {x^{n+1}}{n+1}}+C$ for $n\neq -1$ ,
For setup of the problem we need to integrate the region between the x - axis, the curve, $x=0$ (the y-axis), and $x=2$ .

Solution:

Step 1:
Set up the integral:
$\int _{0}^{2}6x^{2}+2x\,dx.$

Step 2:
Using the power rule we have:
$\int _{0}^{2}6x^{2}+2x\,dx\,=\,6\cdot {\frac {x^{3}}{3}}+2\cdot {\frac {x^{2}}{2}}{\Bigr \|}_{x\,=\,0}^{2}\,=\,2x^{3}+x^{2}{\Bigr \|}_{x\,=\,0}^{2}.$

Step 3:
Finally, we need to evaluate:
$2x^{3}+x^{2}{\Bigr \|}_{x\,=\,0}^{2}=(2(2)^{3}+(2)^{2})-(0+0)=20.$

Final Answer:
$\int _{0}^{\,2}6x^{2}+2x\,dx\,=\,20.$

@@ 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 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, x = 0 (the y-axis), and x = 4.
+|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 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>
-\int _0^2 6x^2+2x \,dx &=& 2x^3+x^2 \Bigr|_0^2 \\
-\end{array}</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>
 |}