Volume of a Wine Glass: Horizontal Slices - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2021
Views:

165

Pages:

2
Author:

Mara Mathis

Volume of a Wine Glass: Horizontal Slices Now we know all of the techniques of integration anyone knows. We’ll celebrate by using our new techniques to answer an interesting question. Find the volume of an exponential wine glass whose bowl is formed by ro tating the portion of the graph of y = ex that joins (0, 1) and (1, e) about the y-axis. Figure 1: A wine glass formed by rotating the graph of y = ex about the y-axis. There are two methods of solving this problem: horizontal and vertical slices. If we compute the volume using horizontal slices we’ll be adding up the volumes of disks with height dy and radius x = ln y. (See Figure 2.) The volume will therefore be: � e � e 2 πx dy = π(ln y)2 dy 1 1 e = πF2 (y)|1 (see previous example) � �e π y(ln y)2 − 2(y ln y − y) 1 � � π (e(ln e)2 − 2(e ln e − e)) − (1(ln 1)2 − 2(1 ln 1 − 1)) � � π (e(1)2 − 2(e · 1 − e)) − (1(0)2 − 2(1 · 0 − 1)) = π [(e − 2(0)) − (−2(−1))] = π(e − 2). = = = Figure 2: Rotating this horizontal slice about the y-axis forms a disk.

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