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Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636
Textbook Problem

(a) Write an expression for a Riemann sum of a function f on an interval [a, b]. Explain the meaning of the notation that you use.

(b) If f ( x ) 0 , what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.

(c) If f(x) takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.

(a)

To determine

To find: The expression for a Riemann sum of a function f.

Answer

The expression for a Riemann sum of a function f is i=1nf(xi*)Δx.

Explanation

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.

R=i=1nf(xi*)Δx

Here, xi* is a point in the i subinterval [xi1,xi] and Δx is the length of the sub intervals.

Thus, the expression for a Riemann sum of a function f is i=1nf(xi*)Δx.

b)

To determine

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

Explanation

Given information:

Consider the condition for the function f(x)0

Explanation:

The function f(x)0 represents that the function is in the first quadrant of the graph.

Sketch the curve f(x) in the first quadrant and then separate the area under the curve with n number approximating rectangles.

Show the curve as in Figure 1.

images

Refer to Figure 1

The function f(x) is positive. Hence the sum of areas of rectangles underneath the curve is the Riemann sum.

Thus, the geometric interpretation of a Riemann sum of f(x)0 is defined.

c)

To determine

To define: The geometric interpretation of a Riemann sum, if the function f(x) takes on both positive and negative values.

Explanation

Given information:

The function f(x) takes on both positive and negative values.

Explanation:

The function f(x) takes on both positive and negative values represents that the function is in the first and fourth quadrant of the graph.

Sketch the curve f(x) in the first and third quadrant and then divide the area under the curve and above the curve with n number approximating rectangles.

Show the curve as in Figure 2.

images

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 f(x) has both positive and negative values.

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