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Chapter 13

Definite Integrals

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Problem 1

Find the area under the curve $y=2 x-x^{2}$ from $x=1$ to $x=2$ with $n=4$ left-endpoint rectangles.

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Problem 2

Find the area under the curve $y=2 x-x^{2}$ from $x=1$ to $x=2$ with $n=4$ right-endpoint rectangles.

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Problem 3

Find the area under the curve $y=2 x-x^{2}$ from $x=1$ to $x=2$ using the Trapezoid Rule with $n=4$ .

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Problem 4

Find the area under the curve $y=2 x-x^{2}$ from $x=1$ to $x=2$ using the Midpoint Formula with $n=4$ .

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Problem 5

Find the area under the curve $y=2 x-x^{2}$ from $x=1$ to $x=2$

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Problem 6

$$\text{ Evaluate }\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x$$

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Problem 7

$$\text{ Evaluate }\int_{1}^{9} 2 x \sqrt{x} d x$$

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Problem 8

$$\text{ Evaluate }\int_{0}^{1}\left(x^{4}-5 x^{3}+3 x^{2}-4 x-6\right) d x$$

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Problem 9

$$\text{ Evaluate }\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin x d x$$

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Problem 10

Suppose we are given the following table of values for $x$ and $g(x)$
$$\begin{array}{|l|l|l|l|l|l|l|}\hline x & {0} & {1} & {3} & {5} & {9} & {14} \\ \hline g(x) & {10} & {8} & {11} & {17} & {20} & {23} \\ \hline\end{array}$$
Use a left-hand Riemann sum with 5 subintervals indicated by the data in the table to approximate $\int_{0}^{14} g(x) d x$

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