Chapter 5.P, Problem 4P

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

A cylindrical glass of radius *r* and height *L* is filled with water and then tilted until the

water remaining in the glass exactly covers its base.

(a) Determine a way to “slice” the water into parallel rectangular cross-sections and then *set up* a definite integral for the volume of the water in the glass.

(b) Determine a way to “slice” the water into parallel cross-sections that are trapezoids and then *set up* a definite integral for the volume of the water.

(c) Find the volume of water in the glass by evaluating one of the integrals in part (a) or part (b).

(d) Find the volume of the water in the glass from purely geometric considerations.

(e) Suppose the glass is tilted until the water exactly covers half the base. In what direction can you “slice” the water into triangular cross-sections? Rectangular cross-sections? Cross-sections that are segments of circles? Find the volume of water in the glass.

To determine

**(a)**

A way to “slice” the water into parallel rectangular cross-sections and thenset upa definite integral for the volume of the water in the glass

Explanation

**Concept:**

Let

**Calculation:**

Consider the diagram below.

In order to take rectangular cross sections, slice the water in

Viewing from FV forms the circle of radius

Equation of the circle passing through the origin:

Now width of rectangle is

To calculate length

To determine

**(b)**

A way to slice water into parallel cross-sections and then set up a definite integral for the volume of the water in the glass

To determine

**(c)**

**To find:**

Volume of water

To determine

**(d)**

**To find:**

Volume of water

To determine

**(e)**

**To find:**

Volume of water

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