About This Chapter
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.
1. Summation Notation and Mathematical Series
The capital Greek letter sigma might be most recognizable as a common symbol in a fraternity or sorority, but it's used for some pretty cool math tricks too. In this lesson, learn how to use sigma for things like determining how many seats are in Michigan Stadium.
2. How to Use Riemann Sums for Functions and Graphs
Find out how Riemann sums can be used to calculate multiple areas efficiently. In this lesson, you'll learn how this can come in handy for irregular areas and how you can put it to use.
3. How to Identify and Draw Left, Right and Middle Riemann Sums
To overestimate or underestimate? In this lesson, you will draw Riemann rectangles so the right side, left side or middle of the rectangle hits the curve.
4. What is the Trapezoid Rule?
In this lesson, you'll move beyond simple rectangles to estimate the area under a curve. Get more sophisticated with your approximations and use trapezoids instead of those pesky rectangles.
5. How to Find the Limits of Riemann Sums
What would happen if you could draw an infinite number of infinitesimally thin rectangles? You'd get the exact area under a curve! Define the Holy Grail of calculus, the integral, in this lesson.
6. Definite Integrals: Definition
Explore how driving backwards takes you where you've already been as we define definite integrals. This lesson will also teach you the relationship between definite integrals and Riemann sums. Then, discover how an integral changes when it is above and below the x-axis.
7. How to Use Riemann Sums to Calculate Integrals
As a new property owner, you might relish mowing your lawn. Up and down your property you mow and measure out small sections to find the area of your property. In this lesson, you will discover what a Riemann sum approach is and how to calculate an estimated area using multiple slices.
8. Linear Properties of Definite Integrals
If you're having integration problems, this lesson will relate integrals to everyday driving examples. We'll review a few linear properties of definite integrals while practicing with some problems.
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Other chapters within the GRE Math: Study Guide & Test Prep course
- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Analytic Geometry in 3-Dimensions
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Determinants & Transformations
- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
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