MATH 53

Midterm 2 | Fall '16 | Steel

1. Evaluate $\int_0^1 \int_0^{\cos^{-1}(x)} e^{\sin y} dy \; dx$. [Recall that for $x \in [0, 1]$ and $y \in [0, \frac{\pi}{2}], \cos^{-1}(x) = y$ iff $\cos(y) = x$.]

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2. Find the centroid of $E$, where $E$ is bounded by the surfaces $x^2 + y^2 = 1, z = x^2 + y^2$, and the plane $z = 0$. [The centroid is the center of mass for a constant density.]

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3. The ellipsoidal shell $E = \{ (x, y, z) | 1 \le \frac{x^2}{4} + \frac{y^2}{9} + z^2 \le 4 \; \text{and} \; z \ge 0 \}$ is filled with matter, with density $d(x, y, z) = z$. Find the total mass.

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4. Let $\vec{F}(x, y) = \frac{1}{x^2 + y^2}< -y, x >$.
(a) Show that $\vec{F}$ is conservative on $D$, where $D = \{ (x, y) | \frac{1}{2} < x \; \text{or} \; \frac{1}{2} < y \}$. What properties does $D$ have that are relevant here?

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(b) Show that $\vec{F}$ is not conservative on $\mathbb{R}^2 - \{ (0, 0) \}$.

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(c) Find $\int_C \vec{F} \cdot d\vec{r}$, where $C$ is any smooth curve in $D$ from $(1, 0)$ to $(0, 4)$. [Hint: Use a convenient one, suggested by the solution of (b).]

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5. True or False. (There is no penalty for guessing wrong.) In the question below, a nice scalar or vector field is one which has partial derivatives of all orders which are continuous everywhere.
a) If $C$ is an oriented smooth curve, and $-C$ is $C$ with the opposite orientation, then $\int_C f \; ds = -\int_{-C} f \; ds$ for any nice scalar field $f$.

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b) If $f$ is continuous on $\mathbb{R}^2$, and $R_a$ is the square with edge-length $a$ centered at $(x_0, y_0)$, then $f(x_0, y_0) = \lim_{a \to 0} \left( \frac{1}{a^2} \int \int_{R_a} f \; dA \right)$

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c) Let $S_1$ and $S_2$ be type I regions in the $uv$-plane, and suppose they are mapped to regions $R_1$ and $R_2$ in the $xy$-plane, respectively, by the transformations $T(u, v) = (e^{u+v}, e^v)$. Then $\frac{\text{area}(R_1)}{\text{area}(S_1)} = \frac{\text{area}(R_2)}{\text{area}(S_2)}$.

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d) The vector field $\vec{F}(\vec{r}) = \frac{\vec{r}}{|\vec{r}|^3}$ on $\mathbb{R}^3 - \{ (0, 0, 0) \}$ is such that its line integrals are independent of path.

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e) If $h$ is a nice scalar field such that $\nabla h = \vec{0}$ everywhere in $\mathbb{R}^3$, then $h$ is constant on $\mathbb{R}^3$.

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f) Let $C$ be a smooth curve in $\mathbb{R}^3$, and suppose $g$ is a nice scalar field such that $g(x, y, z) = k$ for all points $(x, y, z)$ on $C$. Then $\int_C \nabla g \cdot d\vec{r} = 0$.

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