(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
(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
Explanation:
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
Explanation:
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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Student Solutions Manual, Chapters 1-11 for Stewart's Single Variable Calculus, 8th (James Stewart Calculus)
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