# Thomas' Calculus, Early Transcendentals, Books a la Carte Edition (12th Edition) Edit edition Problem 46E from Chapter 12.6: Theory and Examples The barrel shown here is shaped ...

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Chapter: Problem:

Theory and Examples

The barrel shown here is shaped like an ellipsoid with equal pieces cut from the ends by planes perpendicular to the z-axis. The cross-sections perpendicular to the z-axis are circular. The barrel is 2h units high, its midsection radius is R, and its end radii are both r. Find a formula for the barrelâ€™s volume. Then check two things. First, suppose the sides of the barrel are straightened to turn the barrel into a cylinder of radius R and height 2h. Does your formula give the cylinderâ€™s volume? Second, suppose r = 0 and h = R so the barrel is a sphere. Does your formula give the sphereâ€™s volume?

Step-by-step solution:
Chapter: Problem:
• Step 1 of 4

Consider a barrel as shown in Figure 1 which is like an ellipsoid with equal pieces cut from the ends by planes perpendicular to z-axis.

The cross-sections perpendicular to z-axis are circular.

The height of barrel is 2h, the mid-section radius is R and the end radii are r.

(a) The objective is to determine the formula for the volume of the barrel.

(b) Further, the objective is to check if the sides of the barrel are straightened to turn it into a cylinder with radius R and height 2h, if the formula obtained gives volume of the cylinder.

(c) Further, the objective is to check if to make the barrel a sphere, if the formula obtained gives volume of the sphere.

• Chapter , Problem is solved.
Corresponding Textbook

Thomas' Calculus, Early Transcendentals, Books a la Carte Edition | 12th Edition
9780321730787ISBN-13: 032173078XISBN:
This is an alternate ISBN. View the primary ISBN for: Thomas' Calculus Early Transcendentals 12th Edition Textbook Solutions
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