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Slicing the watermelon

Slicing the watermelon

Description:

To demonstrate techniques for approximating and calculating the volume of a simple ellipsoid.

This packet consists of four videos that begin with visualizing the problem geometrically, through approximating the problem and ending with setting up (but not finishing off) the calculus computation for exact volume.

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Tutorial

Introduction

A common problem in Calculus I involves finding the volume of ellipsoid with circular cross-sections in one direction and elliptical cross-sections in another. The easiest physical model for this geometric object is a watermelon.

So let's find the volume of a watermelon that is 28 inches long and 25 inches in diameter. It's a big watermelon.

This packet will set up the problem twice-once as an approximation, the other using precise calculus techniques.

Visualizing the watermelon

This video helps us to visualize the watermelon and the impending slicing process.

Setting up the techniques

This video sets up the paper-and-pencil techniques that are to come and focuses our attention on the important question in each case: "How does the radius of our slice change as we move further towards the end of the watermelon?"

Technique 1: Approximation

This video demonstrates the basic ideas behind approximating the volume of a watermelon by slicing and pretending the slices are perfect cylinders.

Technique 2: Exact computation with integrals

This video sets up (but does not complete) the calculus computation using integrals for the volume of the watermelon.