About This Chapter
Approximation of Definite Integrals - Chapter Summary and Learning Objectives
This chapter returns to Riemann sums and how they might be used in a variety of ways to make approximations. Our expert instructors will walk you through the process of making approximations using increasingly accurate shapes and rectangle widths before discussing the average value theorem. After watching the videos take the assessments to get an idea of which areas you may want to review. The lessons in this chapter include discussions of:
- The Trapezoid Rule
- Riemann sums, including left, right, and middle Riemann sums
- The Average Value Theorem
|What is the Trapezoid Rule||Learn to get more accurate in approximations by using trapezoids, which can closer approach the shape of a curved line, rather than rectangles.|
|How to Use Riemann Sums to Calculate Integrals||Math-by-lawnmower is the topic of this video discussing approximations using a series of rectangles of equal width.|
|How to Identify and Draw Left, Right and Middle Riemann Sums||Sometimes you want your rectangles to be entirely within the area, others times you want to overestimate: learn how to determine at which point the function will intersect with the top of your rectangles.|
|Average Value Theorem||In this video our instructors explain how to find the average value of a function and how to use that to determine an area defined by a curved line.|
1. 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.
2. 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.
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. Average Value Theorem
If you know you've gone 120 miles in 2 hours, you're averaging 60 mph. But what if you know your velocity at every point in time and not how far you've gone? In this lesson, learn how to calculate average values using integrals.
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Other chapters within the AP Calculus BC: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Function Continuity
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Derivative Applications
- Finding Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Applying Integration Techniques
- Understanding Sequences & Series
- Series of Constants
- Taylor Series
- Using a Scientific Calculator for Calculus
- AP Calculus BC Flashcards