{"id":11427,"date":"2021-05-28T12:22:11","date_gmt":"2021-05-28T12:22:11","guid":{"rendered":"http:\/\/www.max-sperling.bplaced.net\/?p=11427"},"modified":"2021-05-28T12:22:11","modified_gmt":"2021-05-28T12:22:11","slug":"exercise-5-geometry-analysis","status":"publish","type":"post","link":"http:\/\/www.max-sperling.bplaced.net\/?p=11427","title":{"rendered":"Math &#8211; Exercise 5 (Geometry, Analysis)"},"content":{"rendered":"<p><strong>Given<\/strong><br \/>\n<center>The semicircle \\(y = \\sqrt{r^2-x^2}\\) with a radius \\(r>0\\).<\/center><\/p>\n<hr>\n<p><strong>Find<\/strong><br \/>\n<center>The volume of the sphere generated by rotating the semicicle around the x-axis.<\/center><\/p>\n<hr>\n<p><strong>Solution<\/strong><br \/>\n<center><br \/>\n\\(<br \/>\n\\begin{array}{l}<br \/>\n  V_{semicircle} &#038;= 2 \\int_0^{r} \\sqrt{r^2-x^2}\\ dx \\\\<br \/>\n  V_{sphere} &#038;= 2 \\pi \\int_0^{r} r^2-x^2\\ dx = 2 \\pi \\left[ r^2x-\\frac{x^3}{3} \\right]_0^r = 2\\pi (r^3-\\frac{r^3}{3}) = 2\\pi \\frac{2r^3}{3} = \\frac{4 \\pi r^3}{3}<br \/>\n\\end{array}<br \/>\n\\)<br \/>\n<\/center><\/p>\n<hr>\n<p><strong>Answer<\/strong><br \/>\n<center>\\(V_{sphere} = \\frac{4 \\pi r^3}{3}\\)<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given The semicircle with a radius . Find The volume of the sphere generated by rotating the semicicle around the<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[47],"tags":[],"_links":{"self":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/11427"}],"collection":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11427"}],"version-history":[{"count":0,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/11427\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11427"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}