In all cases though, the axis must not be interior to the region itself.īe careful! The horizontal axis cases require that the functions be solved for x rather than y. There are also variations of the formula to cover cases in which the axis of revolution is not the y-axis. This observation leads directly to the following version of the Shell Method formula: The height at a typical sample x-value is equal to the difference of the two function values. When the generating region is defined as the area between two functions, then we have to modify the formula somewhat.Ĭonsider the region between two curves, y = f( x) on top and y = g( x) on bottom, between x = a and x = b. Thus the volume is equal to 672π cubic units. Now set up the Shell Method integral and evaluate to find the volume. In addition, we use a = 2 and b = 6 because we have 2 ≤ x ≤ 6. Fortunately, this is exactly what’s pictured in the figure above.įirst identify the dimensions of a typical shell. (This is the easiest case).įinally, after taking the limit as n → ∞ (so that we have infinitely many shells to fill out the solid), we get the exact formula.įind the volume of the solid generated by revolving the region under f( x) = x 2 + 1, where 2 ≤ x ≤ 6, around the y-axis. And suppose that the y-axis as its axis of symmetry. Now suppose we have a solid of revolution with generating region being the area under a function y = f( x) between x = a and x = b. Then the volume is simply length × height × width as in any rectangular solid. In fact, you can think of cutting the shell along its height and “unrolling” it to produce a thin rectangular slab. Note that the volume is simply the circumference (2π r) times the height ( h) times the thickness ( w). Now let’s take a closer look at a single shell.Īs long as the thickness is small enough, the volume of the shell can be approximated by the formula: Making sure to slice in the same direction as the axis of revolution, you will get a clump of nested shells, or thin hollow cylindrical objects. Starting with the smallest cookie cutter and progressing to larger ones, let’s slice through the dough in concentric rings. And you have a set of circular cookie cutters of various sizes. Imagine that your solid is made of cookie dough. However, the Shell Method requires a different kind of slicing. If you wanted to slice perpendicular to the axis of revolution, then you would get slabs that look like thin cylinders ( disks) or cylinders with circles removed ( washers).
SHELL METHOD CALCULUS HOW TO
First we have to decide how to slice the solid.
Suppose you need to find the volume of a solid of revolution.
So you might want to read up before continuing. We defined solids of revolution in a previous article, AP Calculus Review: Disk and Washer Methods. Solids of Revolution and the Shell Methodīriefly, a solid of revolution is the solid formed by revolving a plane region around a fixed axis. In this article, we’ll review the shell method and show how it solves volume problems on the AP Calculus AB/BC exams. Just as in the Disk/Washer Method (see AP Calculus Review: Disk and Washer Methods), the exact answer results from a certain integral. The Shell Method is a technique for finding the volume of a solid of revolution. By Shaun Ault on Ma, UPDATED ON October 24, 2018, in AP
0 kommentar(er)
