{"id":12016,"date":"2018-03-15t22:13:30","date_gmt":"2018-03-16t05:13:30","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=12016"},"modified":"2018-10-24t03:40:53","modified_gmt":"2018-10-24t10:40:53","slug":"ap-calculus-review-shell-method","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-shell-method\/","title":{"rendered":"ap calculus review: shell method"},"content":{"rendered":"

the shell method<\/strong> is a technique for finding the volume of a solid of revolution<\/strong>. just as in the disk\/washer method (see ap calculus review: disk and washer methods<\/a>), the exact answer results from a certain integral. in this article, we’ll review the shell method and show how it solves volume problems on the ap calculus ab\/bc exams.<\/p>\n

solids of revolution and the shell method<\/h2>\n

briefly, a solid of revolution<\/strong> is the solid formed by revolving a plane region around a fixed axis.<\/p>\n

\"this <\/p>\n

we defined solids of revolution in a previous article, ap calculus review: disk and washer methods<\/a>. so you might want to read up before continuing.<\/p>\n

shells, shells, and more shells…<\/h3>\n

suppose you need to find the volume of a solid of revolution. first we have to decide how to slice the solid. if you wanted to slice perpendicular<\/em> to the axis of revolution, then you would get slabs that look like thin cylinders (disks<\/em>) or cylinders with circles removed (washers<\/em>). however, the shell method<\/em> requires a different kind of slicing.<\/p>\n

imagine that your solid is made of cookie dough. and you have a set of circular cookie cutters of various sizes. starting with the smallest cookie cutter and progressing to larger ones, let’s slice through the dough in concentric rings.<\/p>\n

making sure to slice in the same direction as the axis of revolution, you will get a clump of nested shells<\/strong>, or thin hollow cylindrical objects.<\/p>\n

\"nested<\/p>\n

approximating the volume<\/h3>\n

now let’s take a closer look at a single shell.<\/p>\n

\"cylindrical<\/a>.<\/p>\n

as long as the thickness is small enough, the volume of the shell can be approximated<\/em> by the formula:<\/p>\n

v<\/em> = 2πrhw<\/em><\/p>\n

note that the volume is simply the circumference (2πr<\/em>) times the height (h<\/em>) times the thickness (w<\/em>). in fact, you can think of cutting the shell along its height and “unrolling” it to produce a thin rectangular slab. then the volume is simply length<\/em> × height<\/em> × width<\/em> as in any rectangular solid.<\/p>\n

now suppose we have a solid of revolution with generating region being the area under a function y<\/em> = f<\/em>(x<\/em>) between x<\/em> = a<\/em> and x<\/em> = b<\/em>. and suppose that the y<\/em>-axis as its axis of symmetry. (this is the easiest case).<\/p>\n

\"regionf<\/em>(x<\/em>) = x<\/em>2<\/sup> + 1, between 2 and 6, revolved around the y<\/em>-axis, generating a solid of revolution using shell method” width=”300″ height=”300″ class=”size-full wp-image-12021″ \/> <\/p>\n

imagine what happens to a thin vertical strip of the region as it revolves around the y<\/em>-axis. because the axis is also vertical, the strip will sweep out a cylindrical shell. furthermore, we have the following info about each shell:<\/p>\n