{"id":8302,"date":"2016-12-27t16:41:39","date_gmt":"2016-12-28t00:41:39","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=8302"},"modified":"2016-12-28t11:21:19","modified_gmt":"2016-12-28t19:21:19","slug":"integration-ap-calculus","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/integration-ap-calculus\/","title":{"rendered":"what is the integral of f(x)?"},"content":{"rendered":"<p>integration, along with differentiation, is one of the two main operations in calculus.\u00a0 where differentiation is a tool that allows us to examine rates of change, the integral\u00a0is a tool that allows us to add up infinitesimal pieces of a whole.<\/p>\n<h2>more on integration<\/h2>\n<p>it is best to explore integration with an example.\u00a0 imagine we had an irregular shape that we wanted to find the area of.\u00a0 if we divided the shape into regular rectangles, we could add up the area of all the rectangles and find the approximate area of the entire shape.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8307\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.33.15-pm.png\" alt=\"integration ap calculus\" width=\"193\" height=\"213\" \/><\/p>\n<p>if we now took those rectangles and made them thinner and thinner, our approximation of the area of the whole would become more and more accurate.\u00a0 eventually, as the rectangles became infinitely small (and with that, an infinite number of them), we would perfectly be able to find the area of the original shape.\u00a0 this is what the tool of integration allows us to do.\u00a0 it is therefore used to find concepts such as displacement, area, and volume.<\/p>\n<p>integration is very closely related to <a href=\"\/\/www.catharsisit.com\/hs\/ap\/meant-differentiation-mathematics\/\">differentiation<\/a> through the fundamental theorem of calculus.  (this will be discussed in its own article soon). \u00a0in its simplest form, the operation of integration is the opposite of differentiation.\u00a0 it is therefore first introduced in calculus classes in the form of an antiderivative.<\/p>\n<h2>what is an antiderivative?<\/h2>\n<p>the antiderivative of a function is a new function whose derivative brings us back to the original function.\u00a0 for example, if we have a function f(x) = 3x<sup>2<\/sup>, the antiderivative would be f(x) = x<sup>3<\/sup> + c (where c is a constant), because the derivative of f(x) brings up back to our original function.\u00a0 this is also known as an indefinite integral.<\/p>\n<p>the notation for indefinite integrals is the following:<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-8306 aligncenter\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.33.09-pm.png\" alt=\"integration ap calculus\" width=\"130\" height=\"44\" \/><\/p>\n<p>for example:<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-8305 aligncenter\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.33.06-pm.png\" alt=\"integration ap calculus\" width=\"139\" height=\"45\" \/><\/p>\n<p>we should notice that there is a little &#8220;dx&#8221; at the end of our integration.\u00a0 this means that we are taking our integral \u2018with respect to x\u2019.\u00a0 this is the same dx that we see at the top of our differentiation notation.\u00a0 when we study multi-variable calculus, this will become very important.<\/p>\n<p>we should also notice a c at the end of f(x).\u00a0 this represents a constant. \u00a0we have to put this here because we do not, without outside information, know what this constant could be.\u00a0 when we take the derivative of f(x), no matter what c is, it will go away.<\/p>\n<p>&nbsp;<\/p>\n<p>along with the indefinite integral, there is also the definite integral.<\/p>\n<h2>what is a definite integral?<\/h2>\n<p>a definite integral, in two dimensions, gives the area that exists under a curve between two endpoints.<\/p>\n<p>&nbsp;<\/p>\n<p>for example, let us take the function f(x) = -x<sup>2<\/sup>+10 and the end points [-2, 2].\u00a0 we could find the area under this curve using an indefinite integral.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-medium wp-image-8304\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.32.57-pm-300x91.png\" alt=\"integration ap calculus\" width=\"300\" height=\"91\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.32.57-pm-300x91.png 300w, \/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.32.57-pm.png 362w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-medium wp-image-8303\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.32.53-pm-300x229.png\" alt=\"integration ap calculus\" width=\"300\" height=\"229\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.32.53-pm-300x229.png 300w, \/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-7.32.53-pm.png 523w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>integration is a very powerful tool that allows us to solve a wide range of problems.\u00a0 check back soon to see our blog posts on how to determine various integrals, how to use our calculator to solve integration problems, and for practice ap integral problems.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>integration, along with differentiation, is one of the two main operations in calculus and very important to know for the ap calculus exam. click for more!<\/p>\n","protected":false},"author":224,"featured_media":8304,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[259,241,258,257],"ppma_author":[24930],"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>what is the integral of f(x)? 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