# Converting Ellipse Integration Boundaries To Cylindrical Coordinates

I'm having the following integral, and I'm being asked to convert the integration boundaries to cylindrical coordinates. I've figured out that on XY-plane it's an ellipse having the following equation:

$$\frac{(x-\frac{a}{2})^2}{\frac{9a^2}{4}}+\frac{y^2}{\frac{9a^2}{2}} = 1$$

I realize that $0\leq \theta \leq \frac{\pi}{2}$ and $0 \leq z \leq r^2(\theta)$

But I don't seem to be able to reach an expression for $r(\theta)$.

Answer says that $r(\theta) = 2acos(\theta)$.

Any ideas? Thank you.

Lets analyze the bounds in the integral: \begin{align} 0 &\le x \le 2a\\ 0 &\le y \le \sqrt{2ax-x^2}\\ 0 &\le z \le x^2+y^2\\ \end{align} By squaring the second equation, we have $$0\le y^2 \le 2ax-x^2\quad \Rightarrow\quad 0\le (x-a)^2+y^2 \le a^2,$$ which represents a disc of radius $a$, centered at $(a,0)$. This gives us a more clear image of what the bounds on $x$ and $y$ represent (draw the disc and read the bounds): it is the part of this disc in the $y\ge 0$ plane.
The disc $(x-a)^2+y^2 \le a^2$ has polar equation $2a\cos\theta$ (just replace $x$ and $y$ by $r\cos\theta$ and $r\sin\theta$, respectively.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8907093,"math_prob":0.9999597,"size":497,"snap":"2019-43-2019-47","text_gpt3_token_len":164,"char_repetition_ratio":0.13184585,"word_repetition_ratio":0.0,"special_character_ratio":0.3420523,"punctuation_ratio":0.07070707,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000058,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-13T20:41:21Z\",\"WARC-Record-ID\":\"<urn:uuid:1faa8bc7-8510-4e3d-a016-26b3452d305d>\",\"Content-Length\":\"135088\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ed3ca86-6599-41f5-bbfd-da8dd3a42516>\",\"WARC-Concurrent-To\":\"<urn:uuid:46126096-18df-4dff-9bb7-7295d5187b62>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1842216/converting-ellipse-integration-boundaries-to-cylindrical-coordinates\",\"WARC-Payload-Digest\":\"sha1:P3ROPG4O2SXYVOF4SDGHCJCUQRMFYLCQ\",\"WARC-Block-Digest\":\"sha1:XKC2DDOSIVRVSAXCBE7GZAJSIYNKERKA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496667333.2_warc_CC-MAIN-20191113191653-20191113215653-00303.warc.gz\"}"}