Chapter 15, Problem 1RCC

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Suppose *f* is a continuous function defined on a rectangle *R* = [*a*, *b*] × [*c*, *d*].

(a) Write an expression for a double Riemann sum of f. If *f*(*x*, *y*) ≥ 0, what does the sum represent?

(b) Write the definition of

(c) What is the geometric interpretation of
*f*(*x*, *y*) ≥ 0? What if *f* takes on both positive and negative values?

(d) How do you evaluate

(e) What does the Midpoint Rule for double integrals say?

(f) Write an expression for the average value of *f*.

(a)

To determine

**To write:** An expression for a double Riemann sum of the given function.

Explanation

**Given:**

The continuous function

The double integral of *f* over the rectangle *R* is,

Here,

The given continuous function is

The sample points of each rectangle is denoted by

The image value of the sample points under the function

The Riemann sum constants are denoted by *m*, *n*.

The sum mentioned above

(b)

To determine

**To write:** The definition of

Explanation

The double integral can be expressed in terms of double Riemann sum as follows:

The double integral of *f* over the rectangle *R* is,

Here,

The given continuous function is

The sample points of each rectangle is denoted by

The image value of the sample points under the function

The Riemann sum constants are denoted by *m*, *n*.

(c)

To determine

**To write:** The geometric interpretation of

Explanation

When
*xy-*plane and below the given function. The formula for finding this is given above in part (b).

If suppose the given function *f* takes both positive and negative values, then it does not denote the volume exactly. But, it is taken that the volume of the function of the two graphs one above the *xy-*plane and one below the *xy-*plane.

(d)

To determine

**To evaluate:** The value of the double integral

Explanation

Rewrite the indefinite double integral by definite double integral from the equations or inequalities in the given rectangle. Then, as per the rules of integration, integrate it to get the value of the given double integral. That is,

(e)

To determine

**To interpret:** About the Midpoint Rule for double integrals.

Explanation

The double integral,

Here,
*l, b* are the length and breadth of each rectangle.

The given function is

The mid points of each rectangle is denoted by

The Riemann sum constants are denoted by *m*, *n*.

Separate the given region by small rectangles by the method of Riemann sum for the double integrals. Then, pick the sample points from the Midpoint of each rectangle.

(f)

To determine

**To write:** The expression for the average value of *f.*

Explanation

The area of the given rectangle *R* is denoted by

Then,

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