(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 1CC
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
Want to see more full solutions like this?
Chapter 5 Solutions
CALCULUS,EARLY TRANS LCPO
Additional Math Textbook Solutions
Calculus: Early Transcendentals (2nd Edition)
Elementary Statistics: Picturing the World (7th Edition)
Basic College Mathematics
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Introductory Statistics
Mathematics for the Trades: A Guided Approach (11th Edition) (What's New in Trade Math)
- please answer by showing all the dfalowing necessary step DO NOT GIVE ME THE WRONG ANSWER The sides of a cube of ice are melting at a rate of 1 inch per hour. When its volume is 64 cubic inches, at what rate is its volume changing?arrow_forwardSox & Sin (px) dx 0arrow_forward8 L 8 e ipx dxarrow_forward
- Find the Taylor polynomial T³(×) for the function f centered at the number a. f(x) = xe-2x a = 0 T3(x) =arrow_forwardFor each graph in Figure 16, determine whether f (1) is larger or smaller than the slope of the secant line between x = 1 and x = 1 + h for h > 0. Explain your reasoningarrow_forwardPoints z1 and z2 are shown on the graph.z1 is at (4 real,6 imaginary), z2 is at (-5 real, 2 imaginary)Part A: Identify the points in standard form and find the distance between them.Part B: Give the complex conjugate of z2 and explain how to find it geometrically.Part C: Find z2 − z1 geometrically and explain your steps.arrow_forward
- A polar curve is represented by the equation r1 = 7 + 4cos θ.Part A: What type of limaçon is this curve? Justify your answer using the constants in the equation.Part B: Is the curve symmetrical to the polar axis or the line θ = pi/2 Justify your answer algebraically.Part C: What are the two main differences between the graphs of r1 = 7 + 4cos θ and r2 = 4 + 4cos θ?arrow_forwardA curve, described by x2 + y2 + 8x = 0, has a point A at (−4, 4) on the curve.Part A: What are the polar coordinates of A? Give an exact answer.Part B: What is the polar form of the equation? What type of polar curve is this?Part C: What is the directed distance when Ø = 5pi/6 Give an exact answer.arrow_forwardNew folder 10. Find the area enclosed by the loop of the curve (1- t², t-t³)arrow_forward
- 1. Graph and find the corresponding Cartesian equation for: t X== y = t +1 2 te(-∞, ∞) 42,369 I APR 27 F5 3 MacBook Air stv A Aa T 4 DIIarrow_forwardMiddle School GP... Echo home (1) Addition and su... Google Docs Netflix Netflix New folder 9. Find the area enclosed by x = sin²t, y = cost and the y-axis.arrow_forward2. Graph and find the corresponding Cartesian equation for: (4 cos 0,9 sin 0) θ ε [0, 2π) 42,369 I APR 27 3 MacBook Air 2 tv A Aaarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning
- College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning


