# 022 Exam 2 Sample B, Problem 6

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

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

Solution:

Step 1:
Set up the integral:
${\displaystyle \int _{0}^{2}6x^{2}+2x\,dx.}$
Step 2:
Using the power rule we have:
${\displaystyle \int _{0}^{2}6x^{2}+2x\,dx\,=\,6\cdot {\frac {x^{3}}{3}}+2\cdot {\frac {x^{2}}{2}}{\Bigr |}_{x\,=\,0}^{2}\,=\,2x^{3}+x^{2}{\Bigr |}_{x\,=\,0}^{2}.}$
Step 3:
Finally, we need to evaluate:
${\displaystyle 2x^{3}+x^{2}{\Bigr |}_{x\,=\,0}^{2}=(2(2)^{3}+(2)^{2})-(0+0)=20.}$
Final Answer:
${\displaystyle \int _{0}^{\,2}6x^{2}+2x\,dx\,=\,20.}$