022 Exam 2 Sample B, Problem 6

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 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:

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

Step 2:
Using the power rule we have:
${\begin{array}{rcl}\int _{0}^{2}6x^{2}+2x\,dx&=&2x^{3}+x^{2}{\Bigr |}_{0}^{2}\\\end{array}}$

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

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

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