MATH 21B

Final | Fall '16 | Gorsky

(1) Write down the general form of the partial fractions expansion of the rational fucntion given below. You do not have to solve for the constants! $\frac{x^4 + x^3 + x^2 + x + 1}{(x^2 + 1)^2(x^2 - 1)^2(x - 1)(x - 2)^2}$

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(2) Evaluate each of the following integrals.
(a) $\int_{-1}^{1} \frac{1}{x^2} dx$

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(b) $\int \tan^4(x) \sec^4(x) \; dx$

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(c) $\int_1^{\infty} \frac{\ln(x)}{x^2} dx$

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(d) $\int_0^2 | x^2 - 1 | dx$

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(e) $\int x^5 \sqrt{1 - x^2} dx$

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(3) Consider the parametric equations $\begin{align} x(t) = & \cos(t) \\ y(t) = & 2 + \sin(t) \end{align}$ for $0 \le t \le \pi$.
(a) Find the length of the curve.

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(b) Set up but do not evaluate, an integral representing the area of the surface obtained by revolving the curve about the $x$-axis.

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(c) Where does this curve cross the $y$-axis? Give your answer in Cartesian coordinates.

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(4) Set up, but do not evaluate, an integral representing the volume of the solid obtained by revolving the region bounded by $y = x^2 + 2 \qquad \text{and} \qquad x = (y - 2)^2$ about the line $y = 1$.

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(5) Consider the integral $\int_0^\pi 2 \sin(x) dx$ (a) Compute a left hand sum approximation with $n = 2$ subdivisions.

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(b) Compute a midpoint sum approximation with $n = 2$ subdivisions.

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(6) A machine is pulling 100ft of heavy cable up the side of a building. When the cable has been pulled halfway up, the machine malfunctions. Your instructor happens to be passing by, and offers to single-handedly pull the cable the rest of the way up the building. If the cable weighs 10 lb/ft, how much work is needed for your instructor to pull the cable the rest of the way?

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(7) Solve the following initial value problem. $\frac{dy}{dx} = 3x^2e^y \qquad y(0) = -1$

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(8) Determine which of the following statements are true and which are false. Justify your answers.
(a) If $x(t)$ and $y(t)$ are parametric equations and $\frac{dy}{dx} = t^2 + t$, then $\frac{d^y}{dx^2} = 2t + 1$.

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(b) The center of mass of a thin plate must be on the plate.

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(c) For a thin plate whose density function $\delta(x)$ is symmetric about the $y$-axis, the center of mass must have $y$-coordinate $\overline{y} = 0$.

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(d) $\frac{d}{dx} \int_x^2 e^{t^2} dt = e^{x^2}$.

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(Bonus) Find the volume and surface area of the solid obtained by revolving the region bounded by $y = 1/x, x = 1$, and the $x$ axis, about the $x$-axis. Are your answers surprising?

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