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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2:26 minutes

Problem 5.2.31d

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.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Verified step by step guidance

Step 1: Understand the problem. The integral ∫₃⁶ (1 - 2𝓍) d𝓍 represents the area under the curve of the function f(𝓍) = 1 - 2𝓍 from x = 3 to x = 6. We are tasked with determining which Riemann sum (left or right) underestimates or overestimates the value of the definite integral, given n = 6 subintervals.

Step 2: Divide the interval [3, 6] into n = 6 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (b - a) / n, where a = 3 and b = 6. Substitute the values to find Δ𝓍.

Step 3: For the left Riemann sum, use the left endpoints of each subinterval to approximate the integral. The formula for the left Riemann sum is: Sₗ = Σ [f(𝓍ᵢ) * Δ𝓍], where 𝓍ᵢ represents the left endpoint of each subinterval. Compute the left endpoints and set up the summation.

Step 4: For the right Riemann sum, use the right endpoints of each subinterval to approximate the integral. The formula for the right Riemann sum is: Sᵣ = Σ [f(𝓍ᵢ) * Δ𝓍], where 𝓍ᵢ represents the right endpoint of each subinterval. Compute the right endpoints and set up the summation.

Step 5: Analyze the behavior of the function f(𝓍) = 1 - 2𝓍. Since f(𝓍) is a decreasing linear function, the left Riemann sum will underestimate the integral (as it uses lower values of f(𝓍)), while the right Riemann sum will overestimate the integral (as it uses higher values of f(𝓍)).

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Key Concepts

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

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The notation ∫ₐᵇ f(x) dx indicates the integral of f(x) from a to b, providing a numerical value that reflects the net area between the curve and the x-axis.

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by dividing the area under the curve into smaller rectangles. The sum can be calculated using left endpoints, right endpoints, or midpoints of the subintervals. The choice of endpoints affects whether the sum underestimates or overestimates the integral, depending on the function's behavior over the interval.

Underestimation and Overestimation

In the context of Riemann sums, underestimation occurs when the chosen rectangles fall below the curve, while overestimation happens when they extend above it. For a decreasing function, the left Riemann sum will typically overestimate the integral, and the right sum will underestimate it. Conversely, for an increasing function, the left sum will underestimate, and the right sum will overestimate the integral.

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Textbook Question