
(a) Write an expression for a Riemann sum of a function f. 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 find: The expression for a Riemann sum of a function
Answer to Problem 1RCC
The expression for a Riemann sum of a function f is
Explanation of Solution
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 define: The geometric interpretation of a Riemann sum with diagram.
Explanation of Solution
Given information:
Consider the condition for the function
The function
Sketch the curve
Show the curve as in Figure 1.
Refer to Figure 1
The function
Thus, the geometric interpretation of a Riemann sum of
c)

To define: The geometric interpretation of a Riemann sum, if the function
Explanation of Solution
Given information:
The function
The function
Sketch the curve
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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