Chapter 15.1, Problem 3E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Use a Riemann sum with *m* = *n =* 2 to estimate the value of
*R =* [0, 2] × [0, l]. Take the sample points to be upper right corners.

(b) Use the Midpoint Rule to estimate the integral in part (a).

(a)

To determine

**To estimate:** The value of given double integral by taking the sample points from upper right corners over the rectangular region *R.*

Explanation

**Formula used:**

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

Here,

The given function is

The sample points of the lower right corner 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*.

**Given:**

The function is

The rectangular region,

The Riemann sum constants,

**Calculation:**

Plot the given rectangle in the graph and pick the sample points at the upper right corners of each rectangle.

From Figure 1, it is observed that the upper right corners of each rectangle is

Therefore,

Substitute

Substitute

(b)

To determine

**To estimate:** The value of given double integral over the rectangle *R.*

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