{"id":12016,"date":"2018-03-15T22:13:30","date_gmt":"2018-03-16T05:13:30","guid":{"rendered":"https:\/\/magoosh.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":"https:\/\/magoosh.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