\begin{equation*} \renewcommand{\C}{\mathbb{C}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\Q}{\mathbb{Q}} \renewcommand{\N}{\mathbb{N}} \renewcommand{\Z}{\mathbb{Z}} \renewcommand{\bar}{\overline} \renewcommand{\Lin}{\mathrm{Lin}} \renewcommand{\dd}{\mathrm{d}} \end{equation*}

Instructions

Either

  • Record and submit a video of up to 5 minutes in which you walk the viewer through the solution of the problem given.

  • Pair up with someone else in class, record and submit a video of up to 8 minutes in which you walk the viewer through the solution of the problem given.

    In case you are doing with a partner (encouraged) each of you must roughly speak for half of the time.

General ideas

  • Imagine that you are talking to someone that has read the book, but does not know how to actually carry out the solution of the problem.
  • Do NOT read out the mathematical formulas. Make sure they are visible and reference them as "this domain"/ "domain \(\Omega_1\)", "this function" / "function \(f_1(x,y,z)\)"…
  • Clearly state what facts you are using.
  • Do not carry out computations live, but have the main steps on screen.
  • Use visual aids (geogebra/hand-drawn)

Best method to record videos

  • Join your personal Zoom meeting, share screen (your video feed will be in the corner). Start recording. Pause recording when you finish a part of the proof.
  • If doing with a partner, this works really well. Pause the recording half-way, make the other person share the screen, resume recording.
  • Other methods are accepted.

Possible ideas/tips

  • Have a slideshow of your computations Google slides / Powerpoint / Keynote / Libreoffice Impress
  • Include excerpts (cutouts/screenshots) from the book of facts your are using.
  • Start this video by saying: "In this video we study … and show that …" i.e. you can state your claim at the beginning.
  • If you want to post-process the video you can use open-source video editing programs like: Openshot.
  • Prepare additional material like a 3D or 2D graph to aid in communicating your ideas.

Submission

Please submit under "File Drop" in Collab. If you submit earlier than the deadline an attempt will be made to assess your submission earlier and provide feedback.

You will get full marks if you solve the problem and provide an explanation for the solution.

Minor computational mistakes will not negatively impact your result as long as it is clear you understand the methods involved in solving the problem.

You will not be allowed to resubmit this assignment.

Objective

This assignments are in preparation for the second midterm that is going to be an oral exam.

1 Problem

Find the value of the integral

\begin{equation*} \int_{\Omega} f(x,y,z) \dd \mathrm{Vol}(x,y,z). \end{equation*}

The function \(f(x,y,z)\) is given by

\begin{equation*} f(x,y,z)= \frac{1}{(1+x^2+(y-z)^2)^2} \end{equation*}

The domain \(\Omega\subset \R^3\) is given by all points \((x,y,z)\in\R^3\) satisfying

\begin{equation*} \begin{aligned}[t] & -1\leq z\leq 1 \\ & x^2+(y-z)^2\leq z^2 \\ &x+y\geq z \end{aligned} \end{equation*}

You may reference any part of Chap 15 in Ste.

Though the material that is sufficient to solve this problem is in Chap 15.2, 15.3, 15.6, and (optionally 15.7).

2 Hints

You are welcome to come discuss your consideration about the shape of \(\Omega\) and how to split \(f(x,y,z)\) at OH and on Piazza.

  • Help yourself with 3D graphing software to determine the shape of \(\Omega\).

  • For any fixed \(z\) study the cross section \((x,y)\colon (x,y,z)\in\Omega\)

    Show that it looks like the part of the cicle with radius \(z\) centered around \((0,z)\) that lies above the line tilted by \(-\pi/2\) passing through the center.

  • Use symmetries to reduce to studying the integral of
\begin{equation*} \frac{1}{(1+x^2+(y-z)^2)^2} \end{equation*}

on \(\Omega_+=\Omega\cap \{z\gt 0\}\).

  • Use Fubini's theorem in \(3D\) to write
\begin{equation*} \int_{\Omega_+} \frac{1}{(1+x^2+(y-z)^2)^2} \dd \mathrm{Vol}(x,y,z)=\int_{z=0}^{1} \Big(\int_{(x,y)\colon (x,y,z)\in\Omega} \frac{1}{(1+x^2+(y-z)^2)^2} \dd \mathrm{Area}(x,y)\Big)\, \dd z \end{equation*}
  • For any fixed \(z\) use the information about the cross section from above to evaluate the integral
\begin{equation*} \Big(\int_{(x,y)\colon (x,y,z)\in\Omega} \frac{1}{(1+x^2+(y-z)^2)^2} \dd \mathrm{Area}(x,y)\Big) \end{equation*}

in polar coordinates centered around \((0,z)\)