The Mean Value Theorem for Integrals
The Mean Value Theorem for Integrals

{“appState”:{“pageLoadApiCallsStatus”:true},”articleState”:{“article”:{“headers”:{“creationTime”:”2016-03-26T18:31:09+00:00″,”modifiedTime”:”2017-04-21T17:14:38+00:00″,”timestamp”:”2022-09-14T18:18:28+00:00″},”data”:{“breadcrumbs”:[{“name”:”Academics & The Arts”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33662″},”slug”:”academics-the-arts”,”categoryId”:33662},{“name”:”Math”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33720″},”slug”:”math”,”categoryId”:33720},{“name”:”Calculus”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33723″},”slug”:”calculus”,”categoryId”:33723}],”title”:”Using the Mean Value Theorem for Integrals”,”strippedTitle”:”using the mean value theorem for integrals”,”slug”:”using-the-mean-value-theorem-for-integrals”,”canonicalUrl”:””,”seo”:{“metaDescription”:”The Mean Value Theorem for Integrals guarantees that for every definite integral, a rectangle with the same area and width exists. Moreover, if you superimpose “,”noIndex”:0,”noFollow”:0},”content”:”<p>The <i>Mean Value Theorem</i> <i>for Integrals </i>guarantees that for every definite integral, a rectangle with the same area and width exists. Moreover, if you superimpose this rectangle on the definite integral, the top of the rectangle intersects the function. This rectangle, by the way, is called the <i>mean-value rectangle </i>for that definite integral. Its existence allows you to calculate the <i>average value</i> of the definite integral.</p>\r\n<p class=\”Warning\”>Calculus boasts <i>two</i> Mean Value Theorems — one for derivatives and one for integrals. Here, you will look at the Mean Value Theorem for Integrals. You can find out about the Mean Value Theorem for Derivatives in <i>Calculus For Dummies </i>by Mark Ryan (Wiley).</p>\r\n<p>The best way to see how this theorem works is with a visual example: </p>\r\n<div class=\”imageBlock\” style=\”width:432px;\”><img src=\”https://www.dummies.com/wp-content/uploads/312084.image0.jpg\” width=\”432\” height=\”184\” alt=\”A definite integral and its mean-value rectangle have the same width and area.\”/><div class=\”imageCaption\”>A definite integral and its mean-value rectangle have the same width and area.</div></div>\r\n<p>The first graph in the figure shows the region described by the definite integral </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312085.image1.png\” width=\”77\” height=\”52\” alt=\”A definite integral.\”/>\r\n<p>This region obviously has a width of 1, and you can evaluate it easily to show that its area is </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312086.image2.png\” width=\”20\” height=\”37\” alt=\”7/3\”/>\r\n<p>The second graph in the figure shows a rectangle with a width of 1 and an area of </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312087.image3.png\” width=\”20\” height=\”37\” alt=\”Seven thirds\”/>\r\n<p>It should come as no surprise that this rectangle’s height is also </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312088.image4.png\” width=\”20\” height=\”37\” alt=\”seven divided by three\”/>\r\n<p>so the top of this rectangle intersects the original function.</p>\r\n<p>The fact that the top of the mean-value rectangle intersects the function is mostly a matter of common sense. After all, the height of this rectangle represents the average value that the function attains over a given interval. This value must fall someplace between the function’s maximum and minimum values on that interval.</p>\r\n<p>Here’s the formal statement of the Mean Value Theorem for Integrals: If <i>f</i>(<i>x</i>) is a continuous function on the closed interval [<i>a</i><i>,</i> <i>b</i>], then there exists a number <i>c</i> in that interval such that:</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312089.image5.png\” width=\”152\” height=\”53\” alt=\”the Mean Value Theorem for Integrals\”/>\r\n<p>This equation may look complicated, but it’s basically a restatement of this familiar equation for the area of a rectangle:</p>\r\n<p>Area = Height · Width</p>\r\n<p>In other words, start with a definite integral that expresses an area, and then draw a rectangle of equal area with the same width (<i>b</i> – <i>a</i>). The height of that rectangle — <i>f</i>(<i>c</i>)<i> — </i>is such that its top edge intersects the function where <i>x</i> = <i>c</i><i>.</i></p>\r\n<p>The value <i>f</i>(<i>c</i>) is the <i>average value</i> of <i>f</i>(<i>x</i>) over the interval [<i>a</i><i>,</i> <i>b</i>]. You can calculate it by rearranging the equation stated in the theorem:</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312090.image6.png\” width=\”144\” height=\”53\” alt=\”The value f(c) is the average value of f(x) over the interval [a, b]\”/>\r\n<p>For example, here’s a figure that illustrates the definite integral</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312091.image7.png\” width=\”48\” height=\”52\” alt=\”A definite integral\”/>\r\n<p>and its mean-value rectangle. </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312092.image8.jpg\” width=\”432\” height=\”258\” alt=\”Graph that illustrates a definite integral and its mean-value rectangle.\”/>\r\n<p>Now, here’s how you calculate the average value of the shaded area:</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312093.image9.png\” width=\”152\” height=\”205\” alt=\”The process to calculate the average value of the shaded area\”/>\r\n<p>Not surprisingly, the average value of this integral is 30, a value between the function’s minimum of 8 and its maximum of 64.</p>”,”description”:”<p>The <i>Mean Value Theorem</i> <i>for Integrals </i>guarantees that for every definite integral, a rectangle with the same area and width exists. Moreover, if you superimpose this rectangle on the definite integral, the top of the rectangle intersects the function. This rectangle, by the way, is called the <i>mean-value rectangle </i>for that definite integral. Its existence allows you to calculate the <i>average value</i> of the definite integral.</p>\r\n<p class=\”Warning\”>Calculus boasts <i>two</i> Mean Value Theorems — one for derivatives and one for integrals. Here, you will look at the Mean Value Theorem for Integrals. You can find out about the Mean Value Theorem for Derivatives in <i>Calculus For Dummies </i>by Mark Ryan (Wiley).</p>\r\n<p>The best way to see how this theorem works is with a visual example: </p>\r\n<div class=\”imageBlock\” style=\”width:432px;\”><img src=\”https://www.dummies.com/wp-content/uploads/312084.image0.jpg\” width=\”432\” height=\”184\” alt=\”A definite integral and its mean-value rectangle have the same width and area.\”/><div class=\”imageCaption\”>A definite integral and its mean-value rectangle have the same width and area.</div></div>\r\n<p>The first graph in the figure shows the region described by the definite integral </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312085.image1.png\” width=\”77\” height=\”52\” alt=\”A definite integral.\”/>\r\n<p>This region obviously has a width of 1, and you can evaluate it easily to show that its area is </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312086.image2.png\” width=\”20\” height=\”37\” alt=\”7/3\”/>\r\n<p>The second graph in the figure shows a rectangle with a width of 1 and an area of </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312087.image3.png\” width=\”20\” height=\”37\” alt=\”Seven thirds\”/>\r\n<p>It should come as no surprise that this rectangle’s height is also </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312088.image4.png\” width=\”20\” height=\”37\” alt=\”seven divided by three\”/>\r\n<p>so the top of this rectangle intersects the original function.</p>\r\n<p>The fact that the top of the mean-value rectangle intersects the function is mostly a matter of common sense. After all, the height of this rectangle represents the average value that the function attains over a given interval. This value must fall someplace between the function’s maximum and minimum values on that interval.</p>\r\n<p>Here’s the formal statement of the Mean Value Theorem for Integrals: If <i>f</i>(<i>x</i>) is a continuous function on the closed interval [<i>a</i><i>,</i> <i>b</i>], then there exists a number <i>c</i> in that interval such that:</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312089.image5.png\” width=\”152\” height=\”53\” alt=\”the Mean Value Theorem for Integrals\”/>\r\n<p>This equation may look complicated, but it’s basically a restatement of this familiar equation for the area of a rectangle:</p>\r\n<p>Area = Height · Width</p>\r\n<p>In other words, start with a definite integral that expresses an area, and then draw a rectangle of equal area with the same width (<i>b</i> – <i>a</i>). The height of that rectangle — <i>f</i>(<i>c</i>)<i> — </i>is such that its top edge intersects the function where <i>x</i> = <i>c</i><i>.</i></p>\r\n<p>The value <i>f</i>(<i>c</i>) is the <i>average value</i> of <i>f</i>(<i>x</i>) over the interval [<i>a</i><i>,</i> <i>b</i>]. You can calculate it by rearranging the equation stated in the theorem:</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312090.image6.png\” width=\”144\” height=\”53\” alt=\”The value f(c) is the average value of f(x) over the interval [a, b]\”/>\r\n<p>For example, here’s a figure that illustrates the definite integral</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312091.image7.png\” width=\”48\” height=\”52\” alt=\”A definite integral\”/>\r\n<p>and its mean-value rectangle. </p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312092.image8.jpg\” width=\”432\” height=\”258\” alt=\”Graph that illustrates a definite integral and its mean-value rectangle.\”/>\r\n<p>Now, here’s how you calculate the average value of the shaded area:</p>\r\n<img src=\”https://www.dummies.com/wp-content/uploads/312093.image9.png\” width=\”152\” height=\”205\” alt=\”The process to calculate the average value of the shaded area\”/>\r\n<p>Not surprisingly, the average value of this integral is 30, a value between the function’s minimum of 8 and its maximum of 64.</p>”,”blurb”:””,”authors”:[{“authorId”:9399,”name”:”Mark Zegarelli”,”slug”:”mark-zegarelli”,”description”:” <b>Mark Zegarelli</b> is a professional writer with degrees in both English and Math from Rutgers University. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. Along the way, he&#8217;s also paid a few bills doing housecleaning, decorative painting, and (for ten hours) retail sales. He likes writing best, though.”,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/9399″}}],”primaryCategoryTaxonomy”:{“categoryId”:33723,”title”:”Calculus”,”slug”:”calculus”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33723″}},”secondaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”tertiaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”trendingArticles”:null,”inThisArticle”:[],”relatedArticles”:{“fromBook”:[{“articleId”:208670,”title”:”Calculus II For Dummies Cheat Sheet”,”slug”:”calculus-ii-for-dummies-cheat-sheet”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/208670″}},{“articleId”:179236,”title”:”Computing Integrals and Representing Integrals as Functions”,”slug”:”computing-integrals-and-representing-integrals-as-functions”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/179236″}},{“articleId”:179235,”title”:”Drawing with 3-D Cartesian Coordinates”,”slug”:”drawing-with-3-d-cartesian-coordinates”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/179235″}},{“articleId”:179234,”title”:”Evaluating Triple Integrals”,”slug”:”evaluating-triple-integrals”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/179234″}},{“articleId”:179233,”title”:”Find the Area Between Two Functions”,”slug”:”find-the-area-between-two-functions”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/179233″}}],”fromCategory”:[{“articleId”:256336,”title”:”Solve a Difficult Limit Problem Using the Sandwich Method”,”slug”:”solve-a-difficult-limit-problem-using-the-sandwich-method”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/256336″}},{“articleId”:255765,”title”:”Solve Limit Problems on a Calculator Using Graphing Mode”,”slug”:”solve-limit-problems-on-a-calculator-using-graphing-mode”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/255765″}},{“articleId”:255755,”title”:”Solve Limit Problems on a Calculator Using the Arrow-Number”,”slug”:”solve-limit-problems-on-a-calculator-using-the-arrow-number”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/255755″}},{“articleId”:255261,”title”:”Limit and Continuity Graphs: Practice Questions”,”slug”:”limit-and-continuity-graphs-practice-questions”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/255261″}},{“articleId”:255255,”title”:”Use the Vertical Line Test to Identify a Function”,”slug”:”use-the-vertical-line-test-to-identify-a-function”,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/255255″}}]},”hasRelatedBookFromSearch”:false,”relatedBook”:{“bookId”:282046,”slug”:”calculus-ii-for-dummies-2nd-edition”,”isbn”:”9781118161708″,”categoryList”:[“academics-the-arts”,”math”,”calculus”],”amazon”:{“default”:”https://www.amazon.com/gp/product/111816170X/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”ca”:”https://www.amazon.ca/gp/product/111816170X/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”indigo_ca”:”http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/111816170X-item.html&cjsku=978111945484″,”gb”:”https://www.amazon.co.uk/gp/product/111816170X/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”de”:”https://www.amazon.de/gp/product/111816170X/ref=as_li_tl?ie=UTF8&tag=wiley01-20″},”image”:{“src”:”https://www.dummies.com/wp-content/uploads/calculus-ii-for-dummies-2nd-edition-cover-9781118161708-202×255.jpg”,”width”:202,”height”:255},”title”:”Calculus II For Dummies”,”testBankPinActivationLink”:””,”bookOutOfPrint”:false,”authorsInfo”:”<b data-author-id=\”9399\”>Mark Zegarelli</b>, a math tutor and writer with 25 years of professional experience, delights in making technical information crystal clear — and fun — for average readers. He is the author of <i>Logic For Dummies</i> and <i>Basic Math &amp; Pre-Algebra For Dummies</i>.”,”authors”:[{“authorId”:9399,”name”:”Mark Zegarelli”,”slug”:”mark-zegarelli”,”description”:” <b>Mark Zegarelli</b> is a professional writer with degrees in both English and Math from Rutgers University. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. Along the way, he&#8217;s also paid a few bills doing housecleaning, decorative painting, and (for ten hours) retail sales. He likes writing best, though.”,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/9399″}}],”_links”:{“self”:”https://dummies-api.dummies.com/v2/books/”}},”collections”:[],”articleAds”:{“footerAd”:”<div class=\”du-ad-region row\” id=\”article_page_adhesion_ad\”><div class=\”du-ad-unit col-md-12\” data-slot-id=\”article_page_adhesion_ad\” data-refreshed=\”false\” \r\n data-target = \”[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118161708&quot;]}]\” id=\”du-slot-63221af4a92a8\”></div></div>”,”rightAd”:”<div class=\”du-ad-region row\” id=\”article_page_right_ad\”><div class=\”du-ad-unit col-md-12\” data-slot-id=\”article_page_right_ad\” data-refreshed=\”false\” \r\n data-target = \”[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118161708&quot;]}]\” id=\”du-slot-63221af4a97ec\”></div></div>”},”articleType”:{“articleType”:”Articles”,”articleList”:null,”content”:null,”videoInfo”:{“videoId”:null,”name”:null,”accountId”:null,”playerId”:null,”thumbnailUrl”:null,”description”:null,”uploadDate”:null}},”sponsorship”:{“sponsorshipPage”:false,”backgroundImage”:{“src”:null,”width”:0,”height”:0},”brandingLine”:””,”brandingLink”:””,”brandingLogo”:{“src”:null,”width”:0,”height”:0},”sponsorAd”:””,”sponsorEbookTitle”:””,”sponsorEbookLink”:””,”sponsorEbookImage”:{“src”:null,”width”:0,”height”:0}},”primaryLearningPath”:”Advance”,”lifeExpectancy”:null,”lifeExpectancySetFrom”:null,”dummiesForKids”:”no”,”sponsoredContent”:”no”,”adInfo”:””,”adPairKey”:[]},”status”:”publish”,”visibility”:”public”,”articleId”:179189},”articleLoadedStatus”:”success”},”listState”:{“list”:{},”objectTitle”:””,”status”:”initial”,”pageType”:null,”objectId”:null,”page”:1,”sortField”:”time”,”sortOrder”:1,”categoriesIds”:[],”articleTypes”:[],”filterData”:{},”filterDataLoadedStatus”:”initial”,”pageSize”:10},”adsState”:{“pageScripts”:{“headers”:{“timestamp”:”2023-07-17T10:50:01+00:00″},”adsId”:0,”data”:{“scripts”:[{“pages”:[“all”],”location”:”header”,”script”:”<!–Optimizely Script–>\r\n<script src=\”https://cdn.optimizely.com/js/10563184655.js\”></script>”,”enabled”:false},{“pages”:[“all”],”location”:”header”,”script”:”<!– comScore Tag –>\r\n<script>var _comscore = _comscore || [];_comscore.push({ c1: \”2\”, c2: \”15097263\” });(function() {var s = document.createElement(\”script\”), el = document.getElementsByTagName(\”script\”)[0]; s.async = true;s.src = (document.location.protocol == \”https:\” ? \”https://sb\” : \”http://b\”) + \”.scorecardresearch.com/beacon.js\”;el.parentNode.insertBefore(s, el);})();</script><noscript><img src=\”https://sb.scorecardresearch.com/p?c1=2&c2=15097263&cv=2.0&cj=1\” /></noscript>\r\n<!– / comScore Tag –>”,”enabled”:true},{“pages”:[“all”],”location”:”footer”,”script”:”<!–BEGIN QUALTRICS WEBSITE FEEDBACK SNIPPET–>\r\n<script type=’text/javascript’>\r\n(function(){var g=function(e,h,f,g){\r\nthis.get=function(a){for(var a=a+\”=\”,c=document.cookie.split(\”;\”),b=0,e=c.length;b<e;b++){for(var d=c[b];\” \”==d.charAt(0);)d=d.substring(1,d.length);if(0==d.indexOf(a))return d.substring(a.length,d.length)}return null};\r\nthis.set=function(a,c){var b=\”\”,b=new Date;b.setTime(b.getTime()+6048E5);b=\”; expires=\”+b.toGMTString();document.cookie=a+\”=\”+c+b+\”; path=/; \”};\r\nthis.check=function(){var a=this.get(f);if(a)a=a.split(\”:\”);else if(100!=e)\”v\”==h&&(e=Math.random()>=e/100?0:100),a=[h,e,0],this.set(f,a.join(\”:\”));else return!0;var c=a[1];if(100==c)return!0;switch(a[0]){case \”v\”:return!1;case \”r\”:return c=a[2]%Math.floor(100/c),a[2]++,this.set(f,a.join(\”:\”)),!c}return!0};\r\nthis.go=function(){if(this.check()){var a=document.createElement(\”script\”);a.type=\”text/javascript\”;a.src=g;document.body&&document.body.appendChild(a)}};\r\nthis.start=function(){var t=this;\”complete\”!==document.readyState?window.addEventListener?window.addEventListener(\”load\”,function(){t.go()},!1):window.attachEvent&&window.attachEvent(\”onload\”,function(){t.go()}):t.go()};};\r\ntry{(new g(100,\”r\”,\”QSI_S_ZN_5o5yqpvMVjgDOuN\”,\”https://zn5o5yqpvmvjgdoun-wiley.siteintercept.qualtrics.com/SIE/?Q_ZID=ZN_5o5yqpvMVjgDOuN\”)).start()}catch(i){}})();\r\n</script><div id=’ZN_5o5yqpvMVjgDOuN’><!–DO NOT REMOVE-CONTENTS PLACED HERE–></div>\r\n<!–END WEBSITE FEEDBACK SNIPPET–>”,”enabled”:false},{“pages”:[“all”],”location”:”header”,”script”:”<!– Hotjar Tracking Code for http://www.dummies.com –>\r\n<script>\r\n (function(h,o,t,j,a,r){\r\n h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)};\r\n h._hjSettings={hjid:257151,hjsv:6};\r\n a=o.getElementsByTagName(‘head’)[0];\r\n r=o.createElement(‘script’);r.async=1;\r\n r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv;\r\n a.appendChild(r);\r\n })(window,document,’https://static.hotjar.com/c/hotjar-‘,’.js?sv=’);\r\n</script>”,”enabled”:false},{“pages”:[“article”],”location”:”header”,”script”:”<!– //Connect Container: dummies –> <script src=\”//get.s-onetag.com/bffe21a1-6bb8-4928-9449-7beadb468dae/tag.min.js\” async defer></script>”,”enabled”:true},{“pages”:[“homepage”],”location”:”header”,”script”:”<meta name=\”facebook-domain-verification\” content=\”irk8y0irxf718trg3uwwuexg6xpva0\” />”,”enabled”:true},{“pages”:[“homepage”,”article”,”category”,”search”],”location”:”footer”,”script”:”<!– Facebook Pixel Code –>\r\n<noscript>\r\n<img height=\”1\” width=\”1\” src=\”https://www.facebook.com/tr?id=256338321977984&ev=PageView&noscript=1\”/>\r\n</noscript>\r\n<!– End Facebook Pixel Code –>”,”enabled”:true}]}},”pageScriptsLoadedStatus”:”success”},”navigationState”:{“navigationCollections”:[{“collectionId”:287568,”title”:”BYOB (Be Your Own Boss)”,”hasSubCategories”:false,”url”:”/collection/for-the-entry-level-entrepreneur-287568″},{“collectionId”:293237,”title”:”Be a Rad Dad”,”hasSubCategories”:false,”url”:”/collection/be-the-best-dad-293237″},{“collectionId”:295890,”title”:”Career Shifting”,”hasSubCategories”:false,”url”:”/collection/career-shifting-295890″},{“collectionId”:294090,”title”:”Contemplating the Cosmos”,”hasSubCategories”:false,”url”:”/collection/theres-something-about-space-294090″},{“collectionId”:287563,”title”:”For Those Seeking Peace of Mind”,”hasSubCategories”:false,”url”:”/collection/for-those-seeking-peace-of-mind-287563″},{“collectionId”:287570,”title”:”For the Aspiring Aficionado”,”hasSubCategories”:false,”url”:”/collection/for-the-bougielicious-287570″},{“collectionId”:291903,”title”:”For the Budding Cannabis Enthusiast”,”hasSubCategories”:false,”url”:”/collection/for-the-budding-cannabis-enthusiast-291903″},{“collectionId”:291934,”title”:”For the Exam-Season Crammer”,”hasSubCategories”:false,”url”:”/collection/for-the-exam-season-crammer-291934″},{“collectionId”:287569,”title”:”For the Hopeless Romantic”,”hasSubCategories”:false,”url”:”/collection/for-the-hopeless-romantic-287569″},{“collectionId”:296450,”title”:”For the Spring Term Learner”,”hasSubCategories”:false,”url”:”/collection/for-the-spring-term-student-296450″}],”navigationCollectionsLoadedStatus”:”success”,”navigationCategories”:{“books”:{“0”:{“data”:[{“categoryId”:33512,”title”:”Technology”,”hasSubCategories”:true,”url”:”/category/books/technology-33512″},{“categoryId”:33662,”title”:”Academics & The Arts”,”hasSubCategories”:true,”url”:”/category/books/academics-the-arts-33662″},{“categoryId”:33809,”title”:”Home, Auto, & Hobbies”,”hasSubCategories”:true,”url”:”/category/books/home-auto-hobbies-33809″},{“categoryId”:34038,”title”:”Body, Mind, & Spirit”,”hasSubCategories”:true,”url”:”/category/books/body-mind-spirit-34038″},{“categoryId”:34224,”title”:”Business, Careers, & Money”,”hasSubCategories”:true,”url”:”/category/books/business-careers-money-34224″}],”breadcrumbs”:[],”categoryTitle”:”Level 0 Category”,”mainCategoryUrl”:”/category/books/level-0-category-0″}},”articles”:{“0”:{“data”:[{“categoryId”:33512,”title”:”Technology”,”hasSubCategories”:true,”url”:”/category/articles/technology-33512″},{“categoryId”:33662,”title”:”Academics & The Arts”,”hasSubCategories”:true,”url”:”/category/articles/academics-the-arts-33662″},{“categoryId”:33809,”title”:”Home, Auto, & Hobbies”,”hasSubCategories”:true,”url”:”/category/articles/home-auto-hobbies-33809″},{“categoryId”:34038,”title”:”Body, Mind, & Spirit”,”hasSubCategories”:true,”url”:”/category/articles/body-mind-spirit-34038″},{“categoryId”:34224,”title”:”Business, Careers, & Money”,”hasSubCategories”:true,”url”:”/category/articles/business-careers-money-34224″}],”breadcrumbs”:[],”categoryTitle”:”Level 0 Category”,”mainCategoryUrl”:”/category/articles/level-0-category-0″}}},”navigationCategoriesLoadedStatus”:”success”},”searchState”:{“searchList”:[],”searchStatus”:”initial”,”relatedArticlesList”:[],”relatedArticlesStatus”:”initial”},”routeState”:{“name”:”Article3″,”path”:”/article/academics-the-arts/math/calculus/using-the-mean-value-theorem-for-integrals-179189/”,”hash”:””,”query”:{},”params”:{“category1″:”academics-the-arts”,”category2″:”math”,”category3″:”calculus”,”article”:”using-the-mean-value-theorem-for-integrals-179189″},”fullPath”:”/article/academics-the-arts/math/calculus/using-the-mean-value-theorem-for-integrals-179189/”,”meta”:{“routeType”:”article”,”breadcrumbInfo”:{“suffix”:”Articles”,”baseRoute”:”/category/articles”},”prerenderWithAsyncData”:true},”from”:{“name”:null,”path”:”/”,”hash”:””,”query”:{},”params”:{},”fullPath”:”/”,”meta”:{}}},”dropsState”:{“submitEmailResponse”:false,”status”:”initial”},”sfmcState”:{“status”:”initial”},”profileState”:{“auth”:{},”userOptions”:{},”status”:”success”}}

You are watching: Using the Mean Value Theorem for Integrals. Info created by Bút Chì Xanh selection and synthesis along with other related topics.