Math 234 — Final Exam Review

About double & triple integrals

1. Conceptual question. Suppose $\cR$ is a region in three dimensional space that is filled by a gas. Let the density of the gas be $\mu(x, y, z)$, and let the velocity of the gas at the point $(x,y,z)$ be given by $v(x, y, z)$.
   1. Write an integral for the total mass of the gas in the region $\cR$.
   2. Write an integral for the total kinetic energy of the gas in the region $\cR$. (By definition the kinetic energy of an object of mass $m$, moving with velocity $v$, is $\frac12 m v^2$.)
   Answer
2. Polar, Spherical and Cylindrical Coordinates. Compute the following integrals \begin{align*} I_1&=\iint_\cR x^2\; dA, \qquad \cR=\{(x,y) \mid a^2 \le x^2+y^2 \le b^2\} \\ I_2&=\iint_\cR x^2\; dA, \qquad \cR=\{(x,y) \mid a^2 \le x^2+y^2 \le b^2, \; y\ge c|x|\} \\ J_1&=\iiint_\cR x^2\; dA, \qquad \cR=\{(x,y,z) \mid a^2 \le x^2+y^2+z^2 \le b^2\} \\ J_2&=\iiint_\cR x^2\; dA, \qquad \cR=\{(x,y,z) \mid a^2 \le x^2+y^2+z^2 \le b^2,\; z\geq\sqrt{x^2+y^2}\} \\ J_3&=\iiint_\cR x^2\; dA, \qquad \cR=\{(x,y,z) \mid a^2 \le x^2+y^2+z^2 \le b^2,\;y\ge |x|\} \\ K&=\iiint_\cR x^2\; dA, \qquad \cR=\{(x,y,z) \mid a^2 \le x^2+y^2 \le b^2, |z|\le c\} \\ \end{align*} Here $a,b,c>0$ are constants. For each integral you should
   1. draw the region $\cR$,
   2. indicate which type of coordinates you plan to use,
   3. rewrite the integral with an appropriate choice of polar/spherical/cylindrical coordinates,
   4. compute the integral.
   5. write the integral that you would have to compute to find the volume of the region $\cR$, and then compute that integral.
   Some answers
3. Switching the order of integration. Consider the integral \[ I = \int_0^{\infty} \int_0^x e^{-x}\;dy\, dx \]
   1. Write the integral as a double integral \[ I = \iint_\cR e^{-x}\; dA, \] and identify the region $\cR$ over which you have to integrate. (Draw $\cR$).
   2. Compute the integral.
   3. Switch the order of integration, i.e. write the integral as \[ I= \int_\ldots^\ldots \int_{\ldots}^\ldots e^{-x} \; dx\, dy \] and determine the integration bounds “$\ldots$”
   4. Compute the resulting integral.
   Answer
4. A 3D region. Let $\cR$ be the three dimensional region in which $x\ge0$, $y\ge0$, $z\ge0$, and which lies below the plane $2x+3y+z=6$.
   1. Compute the volume of $\cR$
   2. Compute the average value of $y$ on $\cR$

About line integrals

1. Given two points $A(0, 3)$ and $B(4,0)$, let $\cC$ be the line segment from $A$ to $B$. Suppose we also have a constant vector field $\vF(x, y) = 2\ve_1 - \ve_2$
   1. Draw the points, the line segment, and the vector field.
   2. Compute \[ L_1 = \int_{\cC} 3 ds,\qquad L_2 = \int_{\cC} \vF \cdot \vT\; ds,\qquad L_3 = \int_{\cC} \vF \cdot \vN\; ds \] without finding a parametrization of $\cC$, and without finding an antiderivative.
   Answers
2.  

   The distance from the point $A$ to $B$ is 7, the length of the constant vectorfield $\vF$ is $\|\vF\|=3$, and the angle between the line segment $AB$ and the vector $\vF$ is $\theta$. Compute the flux of the vectorfield $\vF$ across the line segment $AB$, and the work done by the force $\vF$ if it acts on a particle that moves from $A$ to $B$. Answers

3. If \[ \vF(x, y) = \begin{pmatrix} P(x, y) \\ Q(x, y) \end{pmatrix}, \] then which of these integrals \begin{align*} A &= \int_{\cC} \vF \cdot \vT\; ds\\ B &= \int_{\cC} \vF \cdot \vN\; ds\\ C &= \int_{\cC} \vF \cdot d\vx \\ D &= \int_{\cC} P(x,y) dx + Q(x,y) dy\\ E &= \int_{\cC} Q(x, y) dx - P(x, y)dy \end{align*} are equal to each other?? Answers
4. 1. State Green's theorem for line integrals.
   2. State Green's theorem for flux integrals.
   3. Suppose that $\cC$ is a closed curve that encloses some domain $\cR$. Let $\vF(x, y) = P(x, y)\ve_1 +Q(x, y)\ve_2$, and suppose $\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0$ holds everywhere in the domain $\cR$. Then \[ \int_\cC \vF\cdot\vN\; ds =0,\quad\text{True or False?} \] Or \[ \int_\cC \vF\cdot\vT\; ds =0,\quad\text{True or False?} \] Answers
5. Let $f(x, y) = xy\cos(x)$, and let $\cC$ be the line segment from the point $(0,0)$ to the point $(\pi, 1)$. Compute \[ \int_\cC \vec\nabla f\cdot\vT\; ds \] Answer