Math 234 — Study guide 3rd midterm

Topics covered on the third midterm

Optimization

• Interior minima and maxima are always critical points.
• Detect critical points given the zero set, or some other level set of a function.
• Know how to find critical points for functions of two or three variables
• Use the second derivative test to decide if a critical point is a min, max, saddle, or some other kind of point. At a saddle point know how to find the tangents to the level set.
• Use Lagrange’s method to find constrained maxima or minima.

Integration

• Compute double and triple integral over rectangles (blocks in 3 dimensions) and other regions.
• Switch the order of integration in double integrals.
• Use polar, cylindrical, and spherical coordinates to compute double or triple integrals.

Review problems

1. Find the largest and smallest value that the function $f(x,y) = x^2-2x+y^2$ has on the disc $D$ given by $x^2+y^2 \le 4$. Note that the domain includes the boundary and that the maximal/minimal values could be attained either in the interior or on the boundary. This is not a short problem: it asks you to think about all optimization methods you have learned this semester.
2. Find the largest and smallest value that the function $f(x,y) = x^2+4y-y^2$ has on the disc $D$ given by $x^2+y^2 \le 1$. Note that again the domain includes the boundary and that the maximal/minimal values could be attained either in the interior or on the boundary.
3. 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
4. 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
5. 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
6. 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 $\cR$