{"id":7881,"date":"2016-09-30t17:29:11","date_gmt":"2016-10-01t00:29:11","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=7881"},"modified":"2016-12-12t18:17:10","modified_gmt":"2016-12-13t02:17:10","slug":"compute-antiderivatives-function-polynomial","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/compute-antiderivatives-function-polynomial\/","title":{"rendered":"ap calculus: computing the antiderivatives of functions and polynomials"},"content":{"rendered":"<p>warm-up: keeping with the theme of volume of solids of revolution, try this multiple choice question, which should need only understanding of the geometric meaning of the definite integral as a measurement of volume.<\/p>\n<p><b>question: <\/b>a solid is generated by revolving the region enclosed by the function <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_0e9628009ff8efdfd32b445b28a7b980.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"y =2sqrt{x}\" title=\"y =2sqrt{x}\"\/>,\u00a0and the lines x =2, x =3, y =1\u00a0about the x-axis. which of the following definite integrals gives the volume of the solid? (hint: draw a picture)<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img1.jpg\"><img decoding=\"async\" class=\"aligncenter size-full wp-image-33\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img1.jpg\" alt=\"ctaoaf_img1\" width=\"146\" height=\"139\" \/><\/a><\/p>\n<h2>calculating the antiderivatives of an integrand<\/h2>\n<p>when we look at an integral <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_974.5_45cd4032c2113f3556cce9e33f97cbfc.png\" style=\"vertical-align:-25.5px; display: inline-block ;\" alt=\"int{a}{b}{f(x) dx}\" title=\"int{a}{b}{f(x) dx}\"\/>we call the values <i>a, b<\/i> the <b>bounds of integration,<\/b> while we call the function <i>f(x)<\/i> the <b>integrand.<\/b> the goal when computing an integral is to first compute an antiderivative <i>f(x)<\/i> for the integrand, for then the fundamental theorem of calculus will allow us to evaluate the right side of\u00a0\u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_976_f1424ad4ecb0ba54d31697f73f93b0a5.png\" style=\"vertical-align:-24px; display: inline-block ;\" alt=\"int{a}{b}f(x)dx = {f(b) - f(a)}\" title=\"int{a}{b}f(x)dx = {f(b) - f(a)}\"\/>,\u00a0which is much easier to evaluate than taking a limit of riemann sums as discussed in the previous post.<\/p>\n<p>therefore we need to learn how to compute antiderivatives in order to learn how to compute integrals.<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img2.jpg\"><img decoding=\"async\" class=\"aligncenter wp-image-34\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img2.jpg\" alt=\"ctaoaf_img2\" width=\"300\" height=\"300\" \/><\/a><\/p>\n<p>the method for computing antiderivatives of a function f(x)\u00a0usually consists recognizing it as the derivative of some function that you know f(x).<\/p>\n<p>we need a little background now, because we are going to develop a toolkit for computing antiderivatives. a <b>set<\/b> is an unordered collection of objects. we can describe the elements of set inside of curly brackets: the integers <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_971.5_909b3c504b5507d51b7c26c8f9b34236.png\" style=\"vertical-align:-28.5px; display: inline-block ;\" alt=\"bbz = {lbrace}..., -2, -1 0, 1, 2, 3, ...{rbrace }\" title=\"bbz = {lbrace}..., -2, -1 0, 1, 2, 3, ...{rbrace }\"\/>,\u00a0, the real numbers <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_300b1a0960abe2e56cdb2200243278b3.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"bbr\" title=\"bbr\"\/>are everything in between. when there are no bounds of integration on an integral, we call\u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_988.5_914335e06659534f1906754ff5c81c3d.png\" style=\"vertical-align:-11.5px; display: inline-block ;\" alt=\"int{ }{ }{f(x) dx}\" title=\"int{ }{ }{f(x) dx}\"\/>an <b>indefinite integral<\/b>. the indefinite integral of a continuous function is the set of antiderivatives \u00a0for the integrand \u00a0so\u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_971.5_49842fb3b08645a890b44aa400602345.png\" style=\"vertical-align:-28.5px; display: inline-block ;\" alt=\"int{ }{ }{f(x) dx} = {lbrace}f:f is an antiderivate for f {rbrace}\" title=\"int{ }{ }{f(x) dx} = {lbrace}f:f is an antiderivate for f {rbrace}\"\/>\u00a0(the set notation is read as \u201cthe set of f such that f is an antiderivative for f.\u201d<\/p>\n<p>so there is more than one antiderivative for a continuous function f? actually, the number of antiderivatives for a continuous function is actually uncountable, and certainly infinite. to see this we need two laws for derivatives: the derivative of a sum of differentiable functions is the sum of the derivatives, and the derivative of a constant function is 0. recall that a function f(x)\u00a0is differentiable if the limit\u00a0<a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img3.jpg\"><img decoding=\"async\" class=\"alignnone size-full wp-image-35\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img3.jpg\" alt=\"ctaoaf_img3\" width=\"95\" height=\"29\" \/><\/a>\u00a0exists for every point x in the domain of f, and we write this value as f'(x).<\/p>\n<p>let f(x), g(x)\u00a0be differentiable functions, then we can see using the definition of the derivative that<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img4.jpg\"><img decoding=\"async\" class=\"size-full wp-image-36 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img4.jpg\" alt=\"ctaoaf_img4\" width=\"514\" height=\"198\" \/><\/a>next, we want to suppose that our function <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_991_355823b21c1f0c7c0c6dc7d670e3410f.png\" style=\"vertical-align:-9px; display: inline-block ;\" alt=\"g(x)= c\" title=\"g(x)= c\"\/> for some\u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_dacb92193c7d5511a53e1fc70fae915d.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"c epsilon bbr\" title=\"c epsilon bbr\"\/>i.e. for some real number c,) then the derivative is zero at every point x:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img5.jpg\"><img decoding=\"async\" class=\"size-full wp-image-37 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img5.jpg\" alt=\"ctaoaf_img5\" width=\"352\" height=\"45\" \/><\/a>now we see why there are so many antiderivatives of an arbitrary continuous function <em>f(x)<\/em>: let <em>f(x)<\/em>\u00a0be any antiderivative of <em>f(x)<\/em>.\u00a0this means that <em>f'(x) = f(x)<\/em>\u00a0for every x in the domain of f. but then <em>f(x) + c\u00a0<\/em>\u00a0where\u00a0<em>c<\/em>\u00a0is a real number is also an antiderivative for\u00a0<em>f ,\u00a0<\/em>since we have that<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img6.jpg\"><img decoding=\"async\" class=\"size-full wp-image-38 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img6.jpg\" alt=\"ctaoaf_img6\" width=\"315\" height=\"39\" \/><\/a><\/p>\n<p>where the first equal sign uses the sum rule for derivatives and the second equal sign uses the fact that the derivative of a constant function is zero.<\/p>\n<p>therefore we write that the indefinite integral of a continuous function is a set of antiderivatives that differ by a real constant:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img7.jpg\"><img decoding=\"async\" class=\"size-full wp-image-39 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoaf_img7.jpg\" alt=\"ctaoaf_img7\" width=\"421\" height=\"28\" \/><\/a><\/p>\n<p><strong>solution to warm-up: a<\/strong><\/p>\n<h2>calculating the antiderivatives of a polynomial<\/h2>\n<p>warm-up:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img1.jpg\"><img decoding=\"async\" class=\"alignnone size-full wp-image-41\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img1.jpg\" alt=\"ctaoap_img1\" width=\"552\" height=\"215\" \/><\/a><\/p>\n<p>(find the solution at the conclusion)<\/p>\n<p>today we want to discuss the computation of the antiderivatives of a polynomial function.\u00a0polynomials, as you might recall, are sums and differences of different powers of the independent variable.\u00a0some functions that are polynomials:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img2.jpg\"><img decoding=\"async\" class=\"alignnone size-full wp-image-42\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img2.jpg\" alt=\"ctaoap_img2\" width=\"252\" height=\"73\" \/><\/a><\/p>\n<p>some functions that are not polynomials:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img3.jpg\"><img decoding=\"async\" class=\"alignnone size-full wp-image-43\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img3.jpg\" alt=\"ctaoap_img3\" width=\"228\" height=\"51\" \/><\/a><\/p>\n<p>notice that h, g, are very similar functions to h, g, so it is important to verify if you are dealing with non-polynomial functions (like <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_991.5_6baccd864912fb859fd43767314e650d.png\" style=\"vertical-align:-8.5px; display: inline-block ;\" alt=\"e^x , log(x), sin(x)\" title=\"e^x , log(x), sin(x)\"\/>)\u00a0or simply constants\u00a0(like <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_a2b2763b7a8c5749654240fa8f062b05.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"e^{37} , log(40), sin(3)\" title=\"e^{37} , log(40), sin(3)\"\/>).\u00a0we now want to understand how to compute the indefinite integral of one of these polynomial functions\u2014which means finding antiderivatives of polynomial functions.<\/p>\n<p>in order to understand antiderivatives, we must first understand derivatives\u2014so the first question we ask is:<\/p>\n<p align=\"center\"><i>what is the derivative of the function f(x) = <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_394cd0838b7e37d98eabbc2ffff473a7.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"x^n\" title=\"x^n\"\/><\/i><i>?<\/i><\/p>\n<p>we can answer this question by computing the limit from the definition:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img4.jpg\"><img decoding=\"async\" class=\"size-full wp-image-44 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img4.jpg\" alt=\"ctaoap_img4\" width=\"358\" height=\"41\" \/><\/a><\/p>\n<p>now what is <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_8bd3a9a7c49ed3d56a9ce5aec851a6a7.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"(x + h)^n\" title=\"(x + h)^n\"\/>?\u00a0if we recall the binomial formula (and pascal\u2019s triangle) we see that\u00a0<a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img5.jpg\"><img decoding=\"async\" class=\"alignnone size-full wp-image-45\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img5.jpg\" alt=\"ctaoap_img5\" width=\"451\" height=\"38\" \/><\/a><\/p>\n<p>substituting this back into the limit we see that:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img6.jpg\"><img decoding=\"async\" class=\"size-full wp-image-46 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img6.jpg\" alt=\"ctaoap_img6\" width=\"447\" height=\"80\" \/><\/a><\/p>\n<p>since all the terms in the numerator have a power of \u201ch\u201d, then we can cancel and are left with:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img7.jpg\"><img decoding=\"async\" class=\"size-full wp-image-47 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img7.jpg\" alt=\"ctaoap_img7\" width=\"244\" height=\"50\" \/><\/a><\/p>\n<p>now since the only term without an \u201ch\u201d is the first term, when <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_901e8f22db15d11d54fe1338292097ce.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"h right 0\" title=\"h right 0\"\/>\u00a0we see that all that we are left with is <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_71d09820ebf61eda8e4e88c8f6d263ce.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"nx^{n-1}\" title=\"nx^{n-1}\"\/>.\u00a0therefore we have that<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img8.jpg\"><img decoding=\"async\" class=\"size-full wp-image-48 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img8.jpg\" alt=\"ctaoap_img8\" width=\"116\" height=\"36\" \/><\/a><\/p>\n<p>we now have figured out the derivatives of all sorts of functions! we have that <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_983.5_ab3ba50e9a5850d79d1055e53985f3fa.png\" style=\"vertical-align:-16.5px; display: inline-block ;\" alt=\"(x^2)\" title=\"(x^2)\"\/>&#8216; = <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_983.5_424cb443effb5616f8f8a4084d0e3706.png\" style=\"vertical-align:-16.5px; display: inline-block ;\" alt=\"2x, (x^3)\" title=\"2x, (x^3)\"\/>&#8216; = <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_e74665e00782113c2a1c05c7d0b35a29.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"3x^{2}\" title=\"3x^{2}\"\/> etc.\u00a0the last pieces of information that we need to know, which i will not prove because it requires advanced structures, is that the integral of a sum of continuous functions is the sum of the integrals. therefore if\u00a0<em>f(x), g(x)<\/em>\u00a0are continuous:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img9.jpg\"><img decoding=\"async\" class=\"size-full wp-image-49 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img9.jpg\" alt=\"ctaoap_img9\" width=\"282\" height=\"35\" \/><\/a><\/p>\n<p>and given a real number \u00a0and a continuous function <em>f(x)<\/em>, the integral of <em>(c.f)(x)\u00a0<\/em>is\u00a0<em>c\u00a0<\/em>times the integral of\u00a0<em>f(x):<\/em><\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img10.jpg\"><img decoding=\"async\" class=\"size-full wp-image-50 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img10.jpg\" alt=\"ctaoap_img10\" width=\"298\" height=\"37\" \/><\/a><\/p>\n<p>this allows us to compute the integral of a sum of polynomial functions, since computing the integral will consist of finding antiderivatives for each term in the sum and then adding them together. this means we only need to know the antiderivative of <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_394cd0838b7e37d98eabbc2ffff473a7.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"x^n\" title=\"x^n\"\/>.\u00a0since a polynomial takes the form <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_985.5_a33b33c9a2a1c288b21a92d618a35d73.png\" style=\"vertical-align:-14.5px; display: inline-block ;\" alt=\"{a_n}{x^n} +{a_{n-1}}{x^{n-1}} + ... + {a_1{x}} + a_0}\" title=\"{a_n}{x^n} +{a_{n-1}}{x^{n-1}} + ... + {a_1{x}} + a_0}\"\/>,\u00a0where the <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_985.5_4bf9c0583cb3cb76b2210aac10703dba.png\" style=\"vertical-align:-14.5px; display: inline-block ;\" alt=\"a_n\" title=\"a_n\"\/>\u00a0is a sequence of real constants, we can compute the integral of one as follows:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img11.jpg\"><img decoding=\"async\" class=\"size-full wp-image-51 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img11.jpg\" alt=\"ctaoap_img11\" width=\"511\" height=\"94\" \/><\/a><\/p>\n<p>so what is the function f that is an antiderivative of \u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_991.5_54a51cab3f374aff6d73a918e1896a68.png\" style=\"vertical-align:-8.5px; display: inline-block ;\" alt=\"f(x)= x^n\" title=\"f(x)= x^n\"\/>?\u00a0\u00a0it will have the form\u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_991.5_1ee5527dbddfea681ad32406cc0431d3.png\" style=\"vertical-align:-8.5px; display: inline-block ;\" alt=\"f(x)= cx^{n+1}\" title=\"f(x)= cx^{n+1}\"\/>,\u00a0and we can solve for \u00a0by differentiating, so that\u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_b7e6356e52b3c3e2dfe8dfc827ca8eeb.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"f(x) = c.(n+1).{x^n}\" title=\"f(x) = c.(n+1).{x^n}\"\/>.\u00a0therefore we have that<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img12a.jpg\"><img decoding=\"async\" class=\"size-full wp-image-55 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img12a.jpg\" alt=\"ctaoap_img12a\" width=\"154\" height=\"94\" \/><\/a><\/p>\n<p>therefore the antiderivatives for\u00a0<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993.5_394cd0838b7e37d98eabbc2ffff473a7.png\" style=\"vertical-align:-6.5px; display: inline-block ;\" alt=\"x^n\" title=\"x^n\"\/>are as follows:<\/p>\n<p><a href=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img13.jpg\"><img decoding=\"async\" class=\"size-full wp-image-54 aligncenter\" src=\"\/\/www.catharsisit.com\/ap\/files\/2013\/10\/ctaoap_img13.jpg\" alt=\"ctaoap_img13\" width=\"158\" height=\"55\" \/><\/a><\/p>\n<p>where <em>c<\/em>\u00a0 is an arbitrary real constant.<\/p>\n<p><strong>warm-up solution: c<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>warm-up: keeping with the theme of volume of solids of revolution, try this multiple choice question, which should need only understanding of the geometric meaning of the definite integral as a measurement of volume. question: a solid is generated by revolving the region enclosed by the function ,\u00a0and the lines x =2, x =3, y [&hellip;]<\/p>\n","protected":false},"author":48,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24929],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>computing the antiderivatives of functions and polynomials<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"\/\/www.catharsisit.com\/hs\/ap\/compute-antiderivatives-function-polynomial\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"ap calculus: computing the antiderivatives of functions and polynomials\" \/>\n<meta property=\"og:description\" content=\"warm-up: keeping 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