022 Exam 2 Sample A, Problem 6

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Find the area under the curve of    between and .

Foundations:  
This problem requires two rules of integration. In particular, you need
Integration by substitution (U - sub): If and are differentiable functions, then

    

The Product Rule: If and are differentiable functions, then

    

The Quotient Rule: If and are differentiable functions and  , then

    
Additionally, we will need our power rule for differentiation:
for ,
as well as the derivative of natural log:

 Solution:

Step 1:  
Set up the integral:
Step 2:  
Using the power rule we have:
Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array}{rcl} \int_1^4 \frac{8}{\sqrt{x}}dx & = & \frac{x^{1/2}} \end{array}}
Step 3:  
Now we need to substitute back into our original variables using our original substitution
to get
Step 4:  
Since this integral is an indefinite integral we have to remember to add "+ C" at the end.
Final Answer:  

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