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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3:03 minutes

Problem 8.8.9

Textbook Question

9. If the Trapezoid Rule is used on the interval [-1, 9] with n = 5 subintervals, at what x-coordinates is the integrand evaluated?

Verified step by step guidance

Step 1: Understand the Trapezoid Rule. The Trapezoid Rule approximates the integral of a function by dividing the interval into 'n' subintervals and calculating the area of trapezoids formed under the curve. The x-coordinates where the integrand is evaluated are the endpoints of these subintervals.

Step 2: Identify the interval and the number of subintervals. The interval is [-1, 9], and the number of subintervals is n = 5.

Step 3: Calculate the width of each subinterval (Δx). The formula for Δx is Δx = (b - a) / n, where 'a' is the starting point of the interval, 'b' is the endpoint, and 'n' is the number of subintervals. Substitute a = -1, b = 9, and n = 5 into the formula.

Step 4: Determine the x-coordinates. Start at the left endpoint (x = -1) and add Δx successively to find the x-coordinates of the subinterval endpoints. These will be x₀, x₁, x₂, x₃, x₄, and x₅.

Step 5: Write the x-coordinates explicitly. The x-coordinates are the points where the integrand is evaluated, and they are evenly spaced based on the calculated Δx.

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Key Concepts

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

Trapezoid Rule

The Trapezoid Rule is a numerical method for approximating the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles, providing a more accurate estimate. The formula involves calculating the average of the function values at the endpoints of each subinterval and multiplying by the width of the subinterval.

Subintervals

In numerical integration, a subinterval is a smaller segment of the overall interval over which the integral is being calculated. The number of subintervals, denoted as 'n', determines how finely the interval is divided, affecting the accuracy of the approximation. More subintervals generally lead to a better approximation of the integral.

Evaluation Points

Evaluation points are the specific x-coordinates at which the function is calculated to apply the Trapezoid Rule. For an interval divided into 'n' subintervals, these points are determined by the endpoints and the width of each subinterval. In this case, the evaluation points will be the endpoints of the interval and the points that divide it into equal segments.

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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

Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.

∫₀² (𝓍²―4) d𝓍

∫₀⁴ (𝓍³―𝓍) d𝓍

3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

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Textbook Question

15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.

15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals

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