Table of contents

0. Functions7h 55m

Introduction to Functions
18m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 5m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

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 31m

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
52m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

13. Intro to Differential Equations2h 55m

Basics of Differential Equations
48m

Slope Fields
19m

Euler's Method
22m

Separable Differential Equations
1h 25m

14. Sequences & Series5h 36m

Sequences
1h 22m

Review of Factorials
11m

Series
44m

Convergence Tests
3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

8. Definite Integrals

Riemann Sums

Calculus

8. Definite Integrals

Riemann Sums

Previous problemNext problem

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).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

0 Comments

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.

Watch next

Master Sigma Notation with a bite sized video explanation from Patrick Ford

Related Videos

Related Practice

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.

views

Textbook Question

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.

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

views

Textbook Question

v = [1 / (2t + 1)] (m/s), for 0 ≤ t ≤ 8 ; n = 4

views

Textbook Question

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

views

Textbook Question

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.