Chapter 15.1, Problem 1E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Estimate the volume of the solid that lies below the surface *z* = *xy* and above the rectangle *R* = {(*x*, *y*) | 0 ≤ *x* ≤ 6, 0 ≤ *y* ≤ 4} Use a Riemann sum with *m =* 3, *n =* 2, and take the sample point to be the upper right corner of each square.

(b) Use the Midpoint Rule to estimate the volume of the solid in part (a).

(a)

To determine

**To estimate:** The volume of the solid lies below the surface and above the rectangle by taking the sample points in the upper right corner of each square.

Explanation

**Formula used:**

The volume of the given solid is,

Here,
*a* is the length of the side of the square.

The sample points of the upper right corner of each square is denoted as

The image value of the sample points under the function *f* is denoted as

And, *m*, *n* is the Riemann sum constants.

**Given:**

The surface,

The rectangle,

The Riemann sum constants,

**Calculation:**

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

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

Therefore,

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

Substitute 2 for *x* and 4 for *y* in the given function

Substitute 4 for *x* and 2 for *y* in the given function

(b)

To determine

**To estimate:** The volume of the solid lies below the surface and above the rectangle by using Midpoint Rule.

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