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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 5, Problem 1RCC

(a)

To determine

**To find:** The expression for a Riemann sum of a function

Expert Solution

The expression for a Riemann sum of a function *f* is

The Riemann sum of a function *f* is the method to find the total area underneath a curve.

The area under the curve dividedas *n* number of approximating rectangles. Hence the Riemann sum of a function *f* is the sum of the area of the all individual rectangles.

Here,

Thus, the expression for a Riemann sum of a function *f* is

b)

To determine

**To define:** The geometric interpretation of a Riemann sum with diagram.

Expert Solution

**Given information:**

Consider the condition for the function

The function

Sketch the curve *n* number approximating rectangles.

Show the curve as in Figure 1.

Refer to Figure 1

The function

Thus, the geometric interpretation of a Riemann sum of

c)

To determine

**To define:** The geometric interpretation of a Riemann sum, if the function

Expert Solution

**Given information:**

The function

The function

Sketch the curve *n* number approximating rectangles.

Show the curve as in Figure 2.

Refer figure 2,

The Riemann sum is the difference of areas of approximating rectangles above and below the *x-*axis

Therefore, the geometric interpretation of a Riemann sum is defined, if

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