# Ch 26: MTLE Mathematics: Area Under the Curve & Integrals

### About This Chapter

## MTLE Mathematics: Area Under the Curve & Integrals - Chapter Summary

In order to master rules and principles that relate to integrals and curves, go through these video lessons. We put them together so you can review:

- Summation notation
- Riemann sums for functions
- Drawing left, right and middle Riemann sums
- The trapezoid rule
- Definite integrals
- Fundamental theorem of calculus and the average value theorem

Working with different theorems and principles relating to integrals doesn't have to be hard if you watch our video lessons. They break down principles in a way that you can understand, getting you ready for the exam quickly. The lessons are accompanied by self-assessment quizzes that help you assess your knowledge and skills.

### 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. 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. The Fundamental Theorem of Calculus

The fundamental theorem of calculus is one of the most important equations in math. In this lesson we start to explore what the ubiquitous FTOC means as we careen down the road at 30 mph.

### 10. 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.

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### Other Chapters

Other chapters within the MTLE Mathematics: Practice & Study Guide course

- MTLE Mathematics: Basic Arithmetic Operations
- MTLE Mathematics: Properties of Real Numbers
- MTLE Mathematics: Operations with Real Numbers
- MTLE Mathematics: Fractions
- MTLE Mathematics: Decimals
- MTLE Mathematics: Percents
- MTLE Mathematics: Ratios, Rates & Proportions
- MTLE Mathematics: Exponents & Exponential Expressions
- MTLE Mathematics: Mathematical Reasoning & Problem Solving
- MTLE Mathematics: Complex Numbers
- MTLE Mathematics: Sequences & Series
- MTLE Mathematics: Functions
- MTLE Mathematics: Algebraic Expressions & Equations
- MTLE Mathematics: Exponential & Logarithmic Functions
- MTLE Mathematics: Vector Operations
- MTLE Mathematics: Matrices in Linear Algebra
- MTLE Mathematics: Linear Functions
- MTLE Mathematics: Linear Inequalities
- MTLE Mathematics: Quadratic Functions
- MTLE Mathematics: Polynomial Functions
- MTLE Mathematics: Rational Expressions
- MTLE Mathematics: Radical Expressions
- MTLE Mathematics: Absolute Value Functions
- MTLE Mathematics: Continuity & Limits
- MTLE Mathematics: Rate of Change & Derivatives
- MTLE Mathematics: Trigonometric Functions
- MTLE Mathematics: Solving Trigonometric Equations
- MTLE Mathematics: Trigonometric Identities
- MTLE Mathematics: Principles of Measurement
- MTLE Mathematics: Area & Perimeter
- MTLE Mathematics: Surface Area & Volume
- MTLE Mathematics: Foundations of Geometry
- MTLE Mathematics: Lines & Angles
- MTLE Mathematics: Properties of Triangles
- MTLE Mathematics: Similar & Congruent Triangle Proofs
- MTLE Mathematics: Right Triangle Proofs
- MTLE Mathematics: Quadrilaterals & Polygons
- MTLE Mathematics: Circles & Arc of a Circle
- MTLE Mathematics: Coordinate Geometry
- MTLE Mathematics: Geometric Transformations
- MTLE Mathematics: Parametric, Polar & Vector Functions
- MTLE Mathematics: Organizing Data
- MTLE Mathematics: Summarizing & Analyzing Data
- MTLE Mathematics: Random Sampling
- MTLE Mathematics: Probability
- MTLE Mathematics: Probability Distributions
- MTLE Mathematics: Discrete Mathematics
- MTLE Mathematics: Mathematical Connections & Communication
- MTLE Mathematics: From Words to Equations, Expressions & Graphs
- MTLE Mathematics: Content Area Reading Skills
- MTLE Mathematics: Instructional Design & Delivery
- MTLE Mathematics: Teaching & Assessing Diverse Learners
- MTLE Mathematics Flashcards