# Practice Problems for Module 12

Section 5.1

Directions. The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos.
1. Estimate the area under the graph of $$f(x)=1+x^2$$ from $$x=-1$$ to $$x=2$$ using the following:
1. three rectangles and right endpoints

$$8$$

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2. three rectangles and left endpoints

$$5$$

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3. three rectangles and midpoints

$$\dfrac{23}{4}$$

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2. Use the definition to find an expression for the area under the graph of $$f$$ as a limit:
$f(x)=\frac{6x}{x^2+8}, \quad 1\leq x \leq 3$

$\lim_{n\rightarrow\infty} \frac{2}{n}\sum_{i=1}^n \frac{6\left(1+\frac{2}{n}i\right)}{\left(1+\frac{2}{n}i\right)^2+8}$

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