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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2:32 minutes

Problem 5.2.32d

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..

∫₀² (𝓍²―2) d𝓍 ; n = 4

Verified step by step guidance

Step 1: Understand the problem. The goal is to approximate the definite integral ∫₀² (𝓍² - 2) d𝓍 using Riemann sums with n = 4 subintervals. Additionally, determine which Riemann sum (left or right) underestimates or overestimates the integral.

Step 2: Divide the interval [0, 2] into n = 4 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (b - a) / n, where a = 0 and b = 2. This gives Δ𝓍 = 2 / 4 = 0.5.

Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function f(𝓍) = 𝓍² - 2. The left endpoints are 𝓍₀ = 0, 𝓍₁ = 0.5, 𝓍₂ = 1, and 𝓍₃ = 1.5. Compute the sum as L = Δ𝓍 * [f(𝓍₀) + f(𝓍₁) + f(𝓍₂) + f(𝓍₃)].

Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function f(𝓍) = 𝓍² - 2. The right endpoints are 𝓍₁ = 0.5, 𝓍₂ = 1, 𝓍₃ = 1.5, and 𝓍₄ = 2. Compute the sum as R = Δ𝓍 * [f(𝓍₁) + f(𝓍₂) + f(𝓍₃) + f(𝓍₄)].

Step 5: Analyze the behavior of the function f(𝓍) = 𝓍² - 2 over the interval [0, 2]. Since the function is increasing on this interval, the left Riemann sum will underestimate the integral (as it uses lower values of the function), and the right Riemann sum will overestimate the integral (as it uses higher values of the function).

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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 value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles provides an estimate of the integral's value. Depending on whether the left or right endpoints of the subintervals are used, the Riemann sum can either overestimate or underestimate the actual area, which is crucial for understanding the behavior of the integral.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ₐᵇ f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over an interval, making it essential for applications in physics, engineering, and economics.

Underestimation and Overestimation

In the context of Riemann sums, underestimation occurs when the sum of the areas of the rectangles is less than the actual area under the curve, while overestimation occurs when the sum exceeds the actual area. For a function that is increasing on the interval, the left Riemann sum will underestimate the integral, and the right Riemann sum will overestimate it. Conversely, for a decreasing function, the opposite is true, highlighting the importance of the function's behavior in determining the accuracy of the approximation.

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Multiple Choice

Express the integral $\int_{0}^{2} 3 x^{2} + 1 d x$ as a limit of Riemann sums using right endpoints. Do not evaluate the limit.

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Textbook Question