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 $n\neq -1$,
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:
$\int _{0}^{\,2}6x^{2}+2x\,dx\,=\,20.$

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