# Find the volume bounded by the paraboloid $x^2+y^2=az$, the cylinder $x^2+y^2=2ay$ and the plane $z=0$

Find the volume bounded by the paraboloid $x^2+y^2=az$, the cylinder $x^2+y^2=2ay$ and the plane $z=0$

My work

Changing to cylindrical coordinates

For Paraboloid

$$x^2+y^2=az\;\;\;\implies r^2=az\;\;\;\implies z=\frac{r^2}{a}$$

For the cylinder

$$x^2+y^2=2ay\;\;\;\implies r=2a \sin \theta$$

Since this volume lies only in first two quadrants $\theta$ goes from $0$ to $\pi$

Volume=$\int_{\theta=0}^{\pi}\int_{r=0}^{2a\sin \theta}\int_{z=0}^{r^2/a}r\ dr\ d\theta \ dz$

• Well, $y\in[0..2a], x\in[-\sqrt{2ay-y^2}..\sqrt{2ay-y^2}], z\in[0..(x^2+y^2)/a]$ but a change of variables might be better. – Graham Kemp Aug 19 '17 at 9:25

Using cylinder coordinate is the best way to solve this problem : $$\left\{(r\cos\theta ,r\sin\theta +a ,z)\mid \theta \in [0,2\pi], r\in[0,a], z\in \left[0,\frac{2r^2+2ar\sin \theta+a^2}{a}\right]\right\}.$$
• @user467745 : Something as $y\in [0,2a]$, $x\in [-\sqrt{2a-(y-1)^2},\sqrt{2a-(y-1)^2}]$ and $z\in [0, \frac{x^2}{a}+\frac{y^2}{a}]$. – Surb Aug 19 '17 at 9:25
• Just to cross check-the answer would be $\pi a^3$, right? – user467745 Aug 19 '17 at 10:27
{"ft_lang_label":"__label__en","ft_lang_prob":0.6338177,"math_prob":0.99999106,"size":468,"snap":"2019-51-2020-05","text_gpt3_token_len":193,"char_repetition_ratio":0.15517241,"word_repetition_ratio":0.0,"special_character_ratio":0.39102563,"punctuation_ratio":0.093457945,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000088,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-22T12:52:53Z\",\"WARC-Record-ID\":\"<urn:uuid:da7db568-75d8-4e0d-a6aa-416d110b9a69>\",\"Content-Length\":\"145593\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2ae5ba98-27e1-4f25-9bbc-11b6e75af46e>\",\"WARC-Concurrent-To\":\"<urn:uuid:ac049b2d-eb39-4b46-953e-6b892527e2d6>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/2398981/find-the-volume-bounded-by-the-paraboloid-x2y2-az-the-cylinder-x2y2-2a\",\"WARC-Payload-Digest\":\"sha1:IPO4XURE4XH6FLB6BTTGAML33VE25P57\",\"WARC-Block-Digest\":\"sha1:I5A4O53K6T63IY3FZBLSFFA2YAHR7FRY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250606975.49_warc_CC-MAIN-20200122101729-20200122130729-00033.warc.gz\"}"}