MATH 53

Midterm 2 | Fall '14 | Agol

1. Find the volume of the solid that lies under the hyperbolic paraboloid $z = 3y^2 - x^2 + 2$ and above the rectangle $R = [-1, 1] \times [1, 2]$ in the $xy$-plane.

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2. Evaluate the integral by reversing the order of integration. $\int_0^{\sqrt{\pi}} \int_y^{\sqrt{\pi}} \cos (x^2) \; dx \; dy$

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3. Let $R$ be the region $R = \{ (x, y) | 1 \le x^2 + y^2 \le 4, 0 \le y \le x \}$. Evaluate the integral by converting to polar coordinates: $\int \int_R \arctan(y/x) \; dA$

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4. Find the volume and centroid of the solid $E$ that lies above the cone $z = \sqrt{x^2 + y^2}$ and below the sphere $x^2 + y^2 + z^2 = 1$, using cylindrical or spherical coordinates, whichever seems more appropriate. [Recall that the centroid is the center of mass of the solid assuming constant density.]

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5. Let $R$ be the parallelogram with vertices $(-1, 3), (1, -3), (3, -1)$ and $(1, 5)$. Use the transformation $x = \frac{1}{4}(u + v), y = \frac{1}{4}(v - 3u)$ to evaluate the integral $\int \int_R (4x + 8y) dA$

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6. Use Green's Theorem to evaluate $\int_C \mathbf{F} \cdot d\mathbf{r}$, where $\mathbf{F} = < e^{-x} + y^2, e^{-y} + x^2 >$, and where $C$ consists of the arc of the curve $y = \cos x$ from $(-\pi/2, 0)$ to $(\pi/2, 0)$ and the line segment from $(\pi/2, 0)$ to $(-\pi/2, 0)$.

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7. Find the curl and divergence of the vector field $\mathbf{F}$. If it is conservative, find a function $f$ such that $\mathbf{F} = \nabla f$. $\mathbf{F}(x, y, z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}}< x, y, z >$

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8. Find the surface area of the surface defined parametrically by the vector equation $\mathbf{r}(u, v) = u\cos v \mathbf{i} + u\sin v \mathbf{j} + v\mathbf{k}, 0 \le u \le 1, 0 \le v \le u$.

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