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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2:55 minutes

Problem 5.1.23

Textbook Question

Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.
ƒ(𝓍) = x + 1 on [1,6] ; n = 5

Verified step by step guidance

Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function ƒ(x) = x + 1 on the interval [1,6] with n = 5 subintervals. The left Riemann sum uses the left endpoints of each subinterval, while the right Riemann sum uses the right endpoints.

Step 2: Determine the width of each subinterval. The interval [1,6] is divided into n = 5 subintervals. The width of each subinterval, Δx, is calculated as Δx = (6 - 1) / 5 = 1.

Step 3: Identify the endpoints of the subintervals. The subintervals are [1,2], [2,3], [3,4], [4,5], and [5,6]. For the left Riemann sum, the left endpoints are x = 1, 2, 3, 4, and 5. For the right Riemann sum, the right endpoints are x = 2, 3, 4, 5, and 6.

Step 4: Calculate the function values at the endpoints. For the left Riemann sum, evaluate ƒ(x) = x + 1 at x = 1, 2, 3, 4, and 5. For the right Riemann sum, evaluate ƒ(x) = x + 1 at x = 2, 3, 4, 5, and 6.

Step 5: Compute the sums. Multiply each function value by the width of the subinterval (Δx = 1) and sum them up. For the left Riemann sum, sum ƒ(1), ƒ(2), ƒ(3), ƒ(4), and ƒ(5). For the right Riemann sum, sum ƒ(2), ƒ(3), ƒ(4), ƒ(5), and ƒ(6).

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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 definite integral of a function over a specified interval. They involve dividing the interval into smaller subintervals, calculating the area of rectangles formed by the function's values at specific points (left, right, or midpoints), and summing these areas. This technique helps in understanding how integrals represent the accumulation of quantities.

Left and Right Riemann Sums

Left and right Riemann sums are specific types of Riemann sums that use the leftmost and rightmost points of each subinterval, respectively, to determine the height of the rectangles. The left Riemann sum uses the function value at the left endpoint of each subinterval, while the right Riemann sum uses the value at the right endpoint. These sums provide different approximations of the area under the curve, with the left sum typically underestimating and the right sum overestimating the integral for increasing functions.

Definite Integral

The definite integral of a function over an interval represents the exact area under the curve of the function between two points. It is calculated as the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral is a fundamental concept in calculus, linking the concepts of area, accumulation, and the fundamental theorem of calculus, which connects differentiation and integration.

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Textbook Question

{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.

ƒ(𝓍) = x² ― 1 on [2,5] ; n = 75

(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.

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Textbook Question

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v = 2t + 1(m/s), for 0 ≤ t ≤ 8 ; n = 2

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{Use of Tech} v = 4 √(t +1) (mi/hr) . for 0 ≤ t ≤ 15 ; n = 5

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Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

f(x) = x + 1 on [0,4]; n = 4

(d) Calculate the left and right Riemann sums.