Chapter 15.1, Problem 2E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

If *R* = [0, 4] × [−1, 2], use a Riemann sum with *m* = 2, *n* = 3 to estimate the value of
*.* Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.

(a)

To determine

**To estimate:** The value of given double integral for the lower 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,

The rectangular region,

The Riemann sum constants,

**Calculation:**

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

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

Therefore,

Substitute 2 for *x* and âˆ’1 for *y* in the given function

Substitute 2 for *x* and 0 for *y* in the given function

Substitute 2 for *x* and 1 for *y* in the given function

Substitute 4 for *x* and âˆ’1 for *y* in the given function

(b)

To determine

**To estimate:** The value of given double integral for the upper left corners over the rectangle *R*.

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