Chapter 5.2, Problem 55E

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

Find the volume of the described solid *S*.

The base of *S* is an elliptical region with boundary curve
*x*-axis are isosceles right triangles with hypotenuse in the base.

To determine

**To find:**

The volume of solid S where base of S is an elliptical region with boundary curve

Explanation

**1) Concept:**

Definition of volume:

Let S be a solid that lies between

**2) Given:**

Equation of base curve is

**3) Calculations:**

The given equation of base curve (ellipse) is

That is

The top half of the ellipse is with a positive square root

Consider a cross section at x, it is an isosceles triangle. The hypotenuse is a chord joining diametrically opposite point of ellipse. From the diagram, it is clear that there is symmetry between upper and lower part of the

Now this is the hypotenuse

By Pythagorean Theorem,

The side length

That is,

Then, the area of isosceles right triangle is

The volume of the solid is the sum of areas of each triangle

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