{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-28T14:38:37+00:00","modifiedTime":"2017-04-18T13:22:17+00:00","timestamp":"2022-09-14T18:18:23+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":"How to Find the Volume of a Solid with a Circular Cross-Section","strippedTitle":"how to find the volume of a solid with a circular cross-section","slug":"how-to-find-the-volume-of-a-solid-with-a-circular-cross-section","canonicalUrl":"","seo":{"metaDescription":"Calculus allows you to calculate the volume of conical objects by dividing the object into an infinite number of circular cross-sections - geometrical shapes re","noIndex":0,"noFollow":0},"content":"<div class='x2 x2-top'><div class=\"video-player-organism\"></div></div>Calculus allows you to calculate the volume of conical objects by dividing the object into an infinite number of circular cross-sections - geometrical shapes resembling pancakes or washers - and adding up the volume of all those cross-sections through integration. This video tutorial shows you how.","description":"<div class='x2 x2-top'><div class=\"video-player-organism\"></div></div>Calculus allows you to calculate the volume of conical objects by dividing the object into an infinite number of circular cross-sections - geometrical shapes resembling pancakes or washers - and adding up the volume of all those cross-sections through integration. This video tutorial shows you how.","blurb":"","authors":[],"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":[],"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":true,"relatedBook":{"bookId":282047,"slug":"calculus-workbook-for-dummies-with-online-practice-3rd-edition","isbn":"9781119357483","categoryList":["academics-the-arts","math","calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119357489/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119357489/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/1119357489-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119357489/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119357489/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/covers/9781119357483.jpg","width":250,"height":350},"title":"Calculus Workbook For Dummies with Online Practice","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><p><b><b data-author-id=\"8957\">Mark Ryan</b> </b>has more than three decades’ experience as a calculus teacher and tutor. He has a gift for mathematics and a gift for explaining it in plain English. He tutors students in all junior high and high school math courses as well as math test prep, and he’s the founder of The Math Center on Chicago’s North Shore. Ryan is the author of <i>Calculus For Dummies, Calculus Essentials For Dummies, Geometry For Dummies</i>, and several other math books.</p>","authors":[{"authorId":8957,"name":"Mark Ryan","slug":"mark-ryan","description":" <p><b>Mark Ryan </b>has more than three decades’ experience as a calculus teacher and tutor. He has a gift for mathematics and a gift for explaining it in plain English. He tutors students in all junior high and high school math courses as well as math test prep, and he’s the founder of The Math Center on Chicago’s North Shore. Ryan is the author of <i>Calculus For Dummies, Calculus Essentials For Dummies, Geometry For Dummies</i>, and several other math books. 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integration. This video tutorial shows you how.","item_vector":null},"titleHighlight":null,"descriptionHighlights":null,"headers":null},{"objectType":"article","id":192156,"data":{"title":"How to Find the Volume of a Shape Using the Washer Method of Integration","slug":"how-to-find-the-volume-of-a-shape-using-the-washer-method-of-integration","update_time":"2016-03-26T21:19:01+00:00","object_type":"article","image":null,"breadcrumbs":[{"name":"Academics & The Arts","slug":"academics-the-arts","categoryId":33662},{"name":"Math","slug":"math","categoryId":33720},{"name":"Calculus","slug":"calculus","categoryId":33723}],"description":"Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. If you have a circular shape with a circular hole in the center, you can use the washer method to find its volume by cutting the shape into thin pieces and then adding up the volumes of the slices. There’s nothing to it.\nHere you go. \n\nand generate a solid by revolving that area about the x-axis.\nA sideways stack of washers — just add up the volumes of all the washers.\nJust think: All the forces of the evolving universe and all the twists and turns of your life have brought you to this moment when you are finally able to calculate the volume of this weird bowl-like shape — something for your diary. So what’s the volume?\n\n Determine where the two curves intersect.\n\nSo the solid in question spans the interval on the x-axis from 0 to 1.\n \n Figure the area of a cross-sectional washer.\n\nThe figure immediately above shows a typical cross section of the 3-D shape, but turned so you’re looking at it head on. Each slice has this washer shape so its area equals the area of the entire circle minus the area of the hole. \nThe area of the circle minus the hole is \n\nwhere R is the outer radius (the big radius) and r is the radius of the hole (the little radius). \n\n \n Multiply this area by the thickness, dx, to get the volume of a representative washer.\n\n \n Add up the volumes of the washers from 0 to 1 by integrating.\n\n \n\nFocus on the simple fact that the area of a washer is the area of the entire disk, \n\nminus the area of the hole, \n\nWhen you integrate, you get \n\nThis is the same, of course, as \n\nwhich is the formula given in most books. But if you just learn that by rote, you may forget it. You’re more likely to remember how to do these problems if you understand the simple big-circle-minus-little-circle idea. ","item_vector":null},"titleHighlight":null,"descriptionHighlights":null,"headers":null},{"objectType":"article","id":179198,"data":{"title":"How to Measure the Volume of an Irregular-Shaped Solid","slug":"how-to-measure-the-volume-of-an-irregular-shaped-solid","update_time":"2016-03-26T18:31:19+00:00","object_type":"article","image":null,"breadcrumbs":[{"name":"Academics & The Arts","slug":"academics-the-arts","categoryId":33662},{"name":"Math","slug":"math","categoryId":33720},{"name":"Calculus","slug":"calculus","categoryId":33723}],"description":"You can measure the volume of any irregular-shaped solid with a cross section that’s a function of x. In some cases, these solids are harder to describe than they are to measure. For example, have a look at this figure.\nA solid based on two exponential curves in space.\nThe solid in the figure consists of two exponential curves — one described by the equation y = ex and the other described by placing the same curve directly in front of the x-axis — joined by straight lines. The other sides of the solid are bounded planes slicing perpendicularly in a variety of directions.\nNotice that when you slice this solid perpendicular with the x-axis, its cross section is always an isosceles right triangle. This is an easy shape to measure, so the slicing method works nicely to measure the volume of this solid. Here are the steps:\n\n Find an expression that represents the area of a random cross section of the solid.\nThe triangle on the y-axis has a height and base of 1 — that is, e0. And the triangle on the line x = 1 has a height and base of e1, which is e. In general, the height and base of any cross section triangle is ex.\nSo here’s how to use the formula for the area of a triangle to find the area of a cross section in terms of x:\n\n \n Use this expression to build a definite integral that represents the volume of the solid.\nNow that you know how to measure the area of a cross section, integrate to add all the cross sections from x = 0 to x = 1:\n\n \n Evaluate this integral to find the volume.\n\n \n","item_vector":null},"titleHighlight":null,"descriptionHighlights":null,"headers":null},{"objectType":"article","id":193540,"data":{"title":"How to Find the Volume of a Shape Using the Washer Method","slug":"how-to-find-the-volume-of-a-shape-using-the-washer-method","update_time":"2016-03-26T21:40:56+00:00","object_type":"article","image":null,"breadcrumbs":[{"name":"Academics & The Arts","slug":"academics-the-arts","categoryId":33662},{"name":"Math","slug":"math","categoryId":33720},{"name":"Calculus","slug":"calculus","categoryId":33723}],"description":"Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. If you have a round shape with a hole in the center, you can use the washer method to find the volume by cutting that shape into thin pieces. Each slice has a hole in its middle that you have to subtract. There’s nothing to it.\nHere you go. \n\nA sideways stack of washers — just add up the volumes of all the washers.\nJust think: All the forces of the evolving universe and all the twists and turns of your life have brought you to this moment when you are finally able to calculate the volume of this solid — something for your diary. So what’s the volume?\n\n Determine where the two curves intersect.\n\nSo the solid in question spans the interval on the x-axis from 0 to 1.\n \n Figure the area of a cross-sectional washer.\n\nIn the above figure, each slice has the shape of a washer so its area equals the area of the entire circle minus the area of the hole. \nThe area of the circle minus the hole is \n\nwhere R is the outer radius (the big radius) and r is the radius of the hole (the little radius). \n\n \n Multiply this area by the thickness, dx, to get the volume of a representative washer.\n\n \n Add up the volumes of the washers from 0 to 1 by integrating.\n\n \n\nFocus on the simple fact that the area of a washer is the area of the entire disk, \n\nminus the area of the hole, \n\nWhen you integrate, you get \n\nThis is the same, of course, as \n\nwhich is the formula given in most books. But if you just learn that by rote, you may forget it. You’re more likely to remember how to do these problems if you understand the simple big-circle-minus-little-circle idea. ","item_vector":null},"titleHighlight":null,"descriptionHighlights":null,"headers":null},{"objectType":"article","id":179196,"data":{"title":"How to Use the Shell Method to Measure the Volume of a Solid","slug":"how-to-use-the-shell-method-to-measure-the-volume-of-a-solid","update_time":"2016-03-26T18:31:16+00:00","object_type":"article","image":null,"breadcrumbs":[{"name":"Academics & The Arts","slug":"academics-the-arts","categoryId":33662},{"name":"Math","slug":"math","categoryId":33720},{"name":"Calculus","slug":"calculus","categoryId":33723}],"description":"The shell method allows you to measure the volume of a solid by measuring the volume of many concentric surfaces of the volume, called “shells.” Although the shell method works only for solids with circular cross sections, it’s ideal for solids of revolution around the y-axis, because you don’t have to use inverses of functions. \nHere’s how it works:\n\n Find an expression that represents the area of a random shell of the solid in terms of x.\n \n Use this expression to build a definite integral (in terms of dx) that represents the volume of the solid.\n \n Evaluate this integral.\n \n\nYou can use a can of soup — or any other can that has a paper label on it — as a handy visual aid to give you insight into how the shell method works. To start out, go to the pantry and get a can of soup.\nSuppose that your can of soup is industrial size, with a radius of 3 inches and a height of 8 inches. You can use the formula for a cylinder to figure out its volume as follows:\nV = Ab · h = 32π · 8 = 72π\nYou can also use the shell method, shown here.\nRemoving the label from a can of soup can help you understand the shell method.\nTo understand the shell method, slice the can’s paper label vertically, and carefully remove it from the can, as shown in the figure. (While you’re at it, take a moment to read the label so that you’re not left with “mystery soup.”)\nNotice that the label is simply a rectangle. Its shorter side is equal in length to the height of the can (8 inches) and its longer side is equal to the circumference (2π · 3 inches = 6π inches). So the area of this rectangle is 48π square inches.\nNow here’s the crucial step: Imagine that the entire can is made up of infinitely many labels wrapped concentrically around each other, all the way to its core. The area of each of these rectangles is:\nA = 2π x · 8 = 16π x\nThe variable x in this case is any possible radius, from 0 (the radius of the circle at the very center of the can) to 3 (the radius of the circle at the outer edge). Here’s how you use the shell method, step by step, to find the volume of the can:\n\n Find an expression that represents the area of a random shell of the can (in terms of x):\nA = 2π x · 8 = 16π x\n \n Use this expression to build a definite integral (in terms of dx) that represents the volume of the can.\nRemember that with the shell method, you’re adding up all the shells from the center (where the radius is 0) to the outer edge (where the radius is 3). So use these numbers as the limits of integration:\n\n \n Evaluate this integral:\n\nNow evaluate this expression:\n= 8π (3)2 – 0 = 72π\nThe shell method verifies that the volume of the can is 72π cubic inches.\n \n","item_vector":null},"titleHighlight":null,"descriptionHighlights":null,"headers":null}]},"relatedArticlesStatus":"success"},"routeState":{"name":"Article3","path":"/article/academics-the-arts/math/calculus/how-to-find-the-volume-of-a-solid-with-a-circular-cross-section-210255/","hash":"","query":{},"params":{"category1":"academics-the-arts","category2":"math","category3":"calculus","article":"how-to-find-the-volume-of-a-solid-with-a-circular-cross-section-210255"},"fullPath":"/article/academics-the-arts/math/calculus/how-to-find-the-volume-of-a-solid-with-a-circular-cross-section-210255/","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"}}# How to Find the Volume of a Solid with a Circular Cross-Section

Calculus allows you to calculate the volume of conical objects by dividing the object into an infinite number of circular cross-sections - geometrical shapes resembling pancakes or washers - and adding up the volume of all those cross-sections through integration. This video tutorial shows you how. ## About This Article

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