# Ch 63: MTEL Math: Area Under the Curve & Integrals

### About This Chapter

## MTEL Math: Area Under the Curve & Integrals - Chapter Summary

The lessons of this chapter will provide you with a review of the different processes involved in finding the area under a curve and integrals. In them, our professional instructors will guide you through these procedures and demonstrate how to solve example problems so that on test day you will be able to accurately answer questions regarding:

- How to identify, draw and use Riemann sums
- Finding the limits of Riemann sums
- The trapezoid rule
- Integrals and calculating integrals
- Identifying the properties of definite integrals
- The average value theorem
- Anti derivatives and indefinite integrals
- Calculating arc lengths of functions

These videos are mobile device-compatible so that if you have access to an Internet-ready mobile device, you won't have to limit your studies to your time in front of a computer. In addition to these lesson videos you may read the lesson transcripts to reinforce your understanding of the information presented in them. If at any time during this chapter you develop a questions regarding one of the topics in these lessons, be sure to ask our professional instructors for assistance via the teacher tabs.

### MTEL Math: Area Under the Curve & Integrals Objectives

Individuals take the Massachusetts Tests for Educator Licensure (MTEL) Mathematics exam to demonstrate that they possess the mathematical skills and knowledge needed to teach math students. During this test, you will be tasked with answering 100 multiple-choice questions and two open-response items in four hours. Sixteen percent of the test will be composed of multiple-choice questions about trigonometry, calculus and discrete mathematics. This fifth subarea of the same name will include a few questions on integral calculus.

### 1. 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.

### 2. 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.

### 3. 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.

### 4. 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.

### 5. 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.

### 6. 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.

### 7. 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.

### 8. 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.

### 9. Indefinite Integrals as Anti Derivatives

What does an anti-derivative have to do with a derivative? Is a definite integral a self-confident version of an indefinite integral? Learn how to define these in this lesson.

### 10. How to Find the Arc Length of a Function

You don't always walk in a straight line. Sometimes, you want to know the distance between two points when the path is curved. In this lesson, you'll learn about finding the length of a curve.

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### Other Chapters

Other chapters within the MTEL Mathematics (09): Practice & Study Guide course

- MTEL Math: Basic Arithmetic Operations
- MTEL Math: Absolute Value & Integers
- MTEL Math: Fractions
- MTEL Math: Decimals
- MTEL Math: Percents
- MTEL Math: Rates & Ratios
- MTEL Math: Proportions
- MTEL Math: Estimation
- MTEL Math: Origins of Math
- MTEL Math: Rational & Irrational Numbers
- MTEL Math: Complex Numbers
- MTEL Math: Properties of Numbers
- MTEL Math: Exponents & Exponential Expressions
- MTEL Math: Roots & Radical Expressions
- MTEL Math: Scientific Notation
- MTEL Math: Number Theory
- MTEL Math: Number Patterns & Sequences
- MTEL Math: Number Patterns & Series
- MTEL Math: Properties of Functions
- MTEL Math: Graphing Functions
- MTEL Math: Factoring
- MTEL Math: The Coordinate Graph & Symmetry
- MTEL Math: Linear Equations
- MTEL Math: Systems of Linear Equations
- MTEL Math: Vectors, Matrices & Determinants
- MTEL Math: Introduction to Quadratics
- MTEL Math: Working with Quadratic Functions
- MTEL Math: Polynomial Functions Basics
- MTEL Math: Higher-Degree Polynomial Functions
- MTEL Math: Piecewise, Absolute Value & Step Functions
- MTEL Math: Rational Expressions, Functions & Graphs
- MTEL Math: Exponential & Logarithmic Functions
- MTEL Math: Measurement
- MTEL Math: Perimeter & Area
- MTEL Math: Polyhedrons & Geometric Solids
- MTEL Math: Symmetry, Similarity & Congruence
- MTEL Math: Properties of Lines
- MTEL Math: Angles
- MTEL Math: Triangles
- MTEL Math: Triangle Theorems & Proofs
- MTEL Math: Similar Polygons
- MTEL Math: The Pythagorean Theorem
- MTEL Math: Quadrilaterals
- MTEL Math: Circles
- MTEL Math: Circular Arcs & Measurement
- MTEL Math: Analytic Geometry & Conic Sections
- MTEL Math: Polar Coordinates & Parameterization
- MTEL Math: Transformations
- MTEL Math: Data & Graphs
- MTEL Math: Statistics
- MTEL Math: Data Collection
- MTEL Math: Samples & Populations
- MTEL Math: Probability
- MTEL Math: Trigonometric Functions
- MTEL Math: Graphs of Trigonometric Functions
- MTEL Math: Trigonometric Identities
- MTEL Math: Applications of Trigonometry
- MTEL Math: Limits
- MTEL Math: Continuity
- MTEL Math: Rate of Change
- MTEL Math: Derivative Calculations & Rules
- MTEL Math: Graphing Derivatives & L'Hopital's Rule
- MTEL Math: Integration Techniques
- MTEL Math: Integration Applications
- MTEL Math: Differential Equations
- MTEL Math: Discrete & Finite Math
- MTEL Mathematics Flashcards