Ch 10: Definite Integrals in Calculus

About This Chapter

Adding all the values in a series, using Riemann sums, and understanding integral basics can be a breeze when you review the lessons in this chapter. Take the quizzes and chapter test to measure your understanding of the material.

Definite Integrals in Calculus - Chapter Summary

The lessons in this chapter help you uncover the basics of definite integrals. You'll see real-world examples covering methods for calculating the area of a property situated on the banks of a curving river and using the Trapezoid Rule to determine the size of the shaded area under a curve in a graph.

The expert instructors in this chapter walk you through the process of using calculus to add up numbers in a series simply. You'll get a refresher in Riemann sums, working first with rectangles then with trapezoids for area estimations before integrals are introduced to find the exact area of various places. After completing these videos you will be able to:

  • Use summation notation to calculate the sum of a series
  • Calculate Riemann sums in functions and graphs
  • Identify left, middle, and right Riemann sums
  • Estimate area using the trapezoid rule
  • Explore limits of Riemann sums for curved-area measurement precision
  • Define integrals and relate them to Riemann sums
  • Understand the linear properties of definite integrals

We bring this calculus topic to you through fun, but informative, videos designed by our professional instructors. Through real-life applications, these videos and their lesson transcripts help you get familiar with integral basics and summation notation. Use the self-assessments after each lesson and at the end of the chapter to see how well you are doing and figure out which lessons may need a second review.

8 Lessons in Chapter 10: Definite Integrals in Calculus
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Summation Notation and Mathematical Series

1. Summation Notation and Mathematical Series

Summation (sigma) notation is the key rule that indicates the sequence of a mathematical series within the start and end terms. Learn more about how summation notation is used to build and break down the formula of a mathematical series.

How to Use Riemann Sums for Functions and Graphs

2. How to Use Riemann Sums for Functions and Graphs

A Riemann sum represents an area that is large and complex as the sum of many smaller, simpler areas. Learn how to use Riemann sums for functions and graphs by using the example of determining the size of a piece of land and using multiple areas.

How to Identify and Draw Left, Right and Middle Riemann Sums

3. How to Identify and Draw Left, Right and Middle Riemann Sums

The Riemann sum is the area between f(x) and the x-axis and it's given by the sum over k=1 to k=n of f(x sub k). Learn more about this mathematical equation, the formula's methodology, and how to draw left, right, and middle Riemann sums.

What is the Trapezoid Rule?

4. What is the Trapezoid Rule?

The trapezoid rule is a way to find the area underneath a curve, using trapezoids to estimate the area, rather than rectangles. Learn more about the trapezoid rule and using trapezoids tot estimate area. Also, review Riemann sum in this lesson.

How to Find the Limits of Riemann Sums

5. How to Find the Limits of Riemann Sums

The area, or Riemann sum, is contained by boundaries, known as limits, that can be calculated. Learn about the concept of Riemann sums, where they are used, and how the limits and integrals can be defined mathematically.

Definite Integrals: Definition

6. Definite Integrals: Definition

A definite integral is found as the limit between a line graphed from an equation, and the x-axis, either positive or negative. Learn how this limit is identified in practical examples of definite integrals.

How to Use Riemann Sums to Calculate Integrals

7. How to Use Riemann Sums to Calculate Integrals

Riemann sums use the method of 'slicing' the area of a graph to isolate the equation used to calculate definite integrals. Follow example problems of using Riemann sums to find an area even when divided into different sections.

Linear Properties of Definite Integrals

8. Linear Properties of Definite Integrals

The linear properties of definite integrals allow complex problems to be solved. Learn how to differentiate between and to use the zero integral property, backward property, constant property, additive property, and sums property.

Chapter Practice Exam
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Practice Final Exam
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