022 Exam 2 Sample A, Problem 6

Find the area under the curve of  $y\,=\,{\frac {8}{\sqrt {x}}}$  between $x=1$ and $x=4$.

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 = 1, and x = 4.

 Solution:

Step 1:
Set up the integral:
$\int _{1}^{4}{\frac {8}{\sqrt {x}}}dx$

Step 2:
Using the power rule we have:
${\begin{array}{rcl}\int _{1}^{4}{\frac {8}{\sqrt {x}}}dx&=&{\frac {8x^{1/2}}{2}}\vert _{1}^{4}\\&=&4x^{1/2}\vert _{1}^{4}\end{array}}$

Step 3:
Now we need to evaluate to get:
$4\cdot 4^{1/2}-4\cdot 1^{1/2}=8-4=4$

Final Answer:
4

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