Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Riemann Sums

Calculus

8. Definite Integrals

Riemann Sums

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1:40 minutes

Problem 5.1.59a

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.

Verified step by step guidance

Step 1: Understand the problem. The question asks whether midpoint Riemann sums give the exact area of the region bounded by the graph of the linear function ƒ(𝓍) = 2𝓍 + 5 and the x-axis on the interval [3,6], regardless of the number of subintervals used.

Step 2: Recall the definition of midpoint Riemann sums. Midpoint Riemann sums approximate the area under a curve by dividing the interval into subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed.

Step 3: Consider the properties of linear functions. Linear functions, such as ƒ(𝓍) = 2𝓍 + 5, are straight lines. The area under a linear function on a closed interval can be calculated exactly using geometry (e.g., the area of a trapezoid) or integration.

Step 4: Analyze the midpoint Riemann sum for linear functions. For linear functions, the midpoint Riemann sum provides the exact area under the curve because the function's rate of change is constant, and the midpoints perfectly capture the average height of the function over each subinterval.

Step 5: Conclude the reasoning. Since the function ƒ(𝓍) = 2𝓍 + 5 is linear, the midpoint Riemann sum will always give the exact area of the region bounded by the graph and the x-axis on the interval [3,6], regardless of the number of subintervals used.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Riemann Sums

Riemann sums are a method for approximating the area under a curve by dividing the region into subintervals and summing the areas of rectangles formed. The height of each rectangle is determined by the function's value at specific points within the subintervals, such as the left endpoint, right endpoint, or midpoint. The accuracy of the approximation improves as the number of subintervals increases, but it may not always yield the exact area unless certain conditions are met.

Midpoint Riemann Sum

A midpoint Riemann sum specifically uses the midpoint of each subinterval to determine the height of the rectangles. This method can provide a better approximation of the area under the curve compared to using left or right endpoints, especially for functions that are not linear. However, for linear functions, like ƒ(𝓍) = 2x + 5, the midpoint Riemann sum will yield the exact area regardless of the number of subintervals used.

Area Under a Curve

The area under a curve represents the integral of a function over a specified interval. For linear functions, the area can be calculated using geometric formulas, such as the area of a trapezoid or triangle, since the graph forms straight lines. In the case of the function ƒ(𝓍) = 2x + 5, the area between the graph and the x-axis on the interval [3,6] can be computed directly, confirming that Riemann sums will yield the exact area due to the linearity of the function.

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Related Practice

Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.

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Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(c) For an increasing or decreasing nonconstant function on an interval [a,b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].

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Textbook Question

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(a) Evaluate the right Riemann sum for the integral with n = 3 .

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Textbook Question

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .