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2nd Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781133112280

Chapter 12, Problem 1RCC

(a)

To determine

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

Expert Solution

The expression for a double Riemann sum is

Given that the continuous function

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

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 *m*, *n*.

The sum mentioned above

(b)

To determine

**To write:** The definition of

Expert Solution

The definition of

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 *m*, *n*.

Thus, the definition of

(c)

To determine

**To write:** The geometric interpretation of

Expert Solution

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

Expert Solution

The value of

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,

Thus, the value of

(e)

To determine

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

Expert Solution

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.

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