MATH 16B

Final | Fall '16

1. Find the following integrals.
(a) $\int \left( \frac{1}{x^2} - x^2 - \frac{1}{3} \right) dx$

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(b) $\int (r^2 + r + 1) e^r \; dr$

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2. Suppose that $f$ and $g$ are continuous functions and $\int_1^5 f(x) dx = -1, \; \int_5^7 f(x) dx = 2, \quad \int_1^7 g(x) dx = 3$ Use the properties of definite integrals to find
(a) $\int_1^7 f(x) dx$

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(b) $\int_7^5 f(x) dx$

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(c) $\int_1^7 (3f(x) - 2g(x)) dx$

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3. Find the following definite integrals.
(a) $\int_{-1}^{-\frac{1}{2}} t^{-2} \sin \left( 1 + \frac{1}{t} \right) dt$

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(b) $\int_1^2 \frac{e^{2 \ln x}}{x} dx$ (Hint: simplify the integrand)

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4. Use partial fractions to find the following integral. $\int \frac{x^2 + 1}{(x^2 - 1)(x + 1)} dx$

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5. Find the volume of the solid generated by revolving the region bounded by $y = 2 - x^2$ and the line $y = 1$ and $x = -1$ about the $x$-axis using disc/washer method.


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6. Let $f(x) = kx^{\frac{3}{2}}$ be a function on the interval $[0, 1]$, where $k$ is a positive constant.
(a) Find $k$ for which $f(x)$ will be a probability density function of some random variable.

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(b) Set up an integral for the expectation random variable. You don't need to evaluate it.

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(c) Set up an integral for the variance of the random variable. You don't have to evaluate it.

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7. Consider the integral $\int_{-1}^{2} \sin x \; dx$.
(a) Use Simpson's Rule with $n = 8$ to approximate the integral. You do not have to simplify the answer.

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(b) If you wanted to use the trapezoid rule to approximate this integral with an error of less than $0.01$, how many trapezoids would you need? The error $E$ in approximating $\int_a^b f(x) dx$ using the trapezoid rule is $|E| \le \frac{(b - a)^3}{12n^2} \left[ \max_{a \le x \le b} | f''(x) | \right]$

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