# Problem 1. Double Integrals

(a) (2 points) Evaluate $\int_0^\pi \int_x^\pi \frac{\sin y}{y} dy \; dx$

(b) (3 points) A thinp late covers the region $R$ bounded by $y = x$ and $y^2 = x$ in the first quadrant. Assume the plate has constant density $\delta$ and mass $M = 1$. Find $delta$.

(c) (3 points) Change to polar coordinates but do not evaluate. $\int_0^2 \int_{-\sqrt{1 - (y - 1)^2}}^{0} xy^2 \; dx \; dy$

# Problem 2. Rectangular, Cylindrical and Spherical coordinates

(a) Set-up, but do not evaluate, volume integrals for the following solids.
(i) (2 points) Cylindrical coordinates. $D$ is the right circular cylinder whose base is the circle $r = 3 \cos \theta$ in the $xy$-plane and whose top lies in the plane $z = 5 - x$.

(ii) (2 points) Spherical coordinates. $D$ lies above the cone $\phi = 3\pi/4$ and between the spheres $x^2 + y^2 + z^2 = 9$ and $x^2 + y^2 + z^2 = 4$.

(b) (3 points) Verify that for any real number $a > 0$ the rectangular equation $az = \sqrt{x^2 + y^2}$ is equivalent to the spherical equation $\phi = \arctan(a)$.

# Problem 3. Curvature

The curvature $κ$ of a smooth curve $C$ is defined as the magnitude of the derivative of the curve's unit tangent vector $\mathbf{T}$ with respect to arc length $s$.
(a) (2 points) For a smooth parametrized curve $C: \mathbf{r}(t), \quad a \le t \le b$ state the formula for calculating $κ$.

(b) (4 points) Compute the curvature function $κ = κ(t)$ for the curve $\mathbf{r}(t) = (e^t \cos t)\mathbf{i} + (e^t \sin t) \mathbf{j} + \mathbf{k}$

# Problem 4. Generalizations

(a) (1 points) The line integral $\int_C f(x, y, z) ds$ generalizes the real line integral $\int_a^b f(x) dx$. Consider a paramterized curve $C: \mathbf{r}(t) = < f(t), g(t), h(t) >, \quad a \le t \le b$ Write the equation relating $ds$ and $dt$.

(b) (3 points) Consider the surface integral $\int \int_S G(x, y, z) d\sigma$ where $S: \mathbf{r}(u, v) = < f(u, v), g(u, v), h(u, v) >, \quad a \le u \le b, c \le v \le d$ Write the surface integral as a paramteric surface integral.

(c) (4 points) Green's Theorem relates certain line integrals to surface integrals. Use Green's Theorem to write flux and circulation as double integrals in del notation.
Flux across $C =$
Circulation around $C =$

(d) (4 points) The Divergence Theorem and Stokes' Theorem generalize Green's Theorem as:
Flux of curl =
Flux across $S =$

# Problem 5. The del operator $\nabla$

Let all partial derivatives of the scalar function $f = f(x, y, z)$ be continuous.
(a) (2 points) Write $\nabla$ as a vector.

(b) (2 points) Write the vector produced by applying the del operator to $f$.

(c) (2 points) Suppose $f$ is a potential function of a vector field $\mathbf{F}$. Then in terms of $f$
$\mathbf{F} =$ and
$\int_A^B \mathbf{F} \cdot d\mathbf{r} =$