# Ch 45: MTTC Math (Secondary): Area Under the Curve & Integrals

### About This Chapter

## MTTC Math (Secondary): Area Under the Curve & Integrals - Chapter Summary

This chapter has been put together to prepare you for questions on the MTTC Mathematics (Secondary) exam asking about finding the area under the curve and calculating integrals. Watch this series of short lesson videos to improve your understanding of:

- Using Riemann sums
- Identifying left, right and middle Riemann sums
- The trapezoid rule
- Finding limits of Riemann sums
- Calculating integrals with Riemann sums
- Linear properties of definite integrals
- Average value theorem
- Finding arc lengths of functions

These lesson videos are taught by professional instructors who are available to answer any questions you may develop as you complete these lessons. Improve your retention of the information presented in these videos by completing the assessments that accompany them. Also use your results from these assessments to find areas of weakness. Fortify these areas by returning to the lesson videos with video tags or lesson transcripts.

### 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 MTTC Mathematics (Secondary) (022): Practice & Study Guide course

- MTTC Math (Secondary): Reasoning & Problem Solving
- MTTC Math (Secondary): Basic Arithmetic
- MTTC Math (Secondary): Absolute Value
- MTTC Math (Secondary): Integers
- MTTC Math (Secondary): Introduction to Fractions
- MTTC Math (Secondary): Operations with Fractions
- MTTC Math (Secondary): Introduction to Decimals
- MTTC Math (Secondary): Operations with Decimals
- MTTC Math (Secondary): Percents
- MTTC Math (Secondary): Estimating & Rounding
- MTTC Math (Secondary): Rational & Irrational Numbers
- MTTC Math (Secondary): Complex Numbers
- MTTC Math (Secondary): Properties of Real Numbers
- MTTC Math (Secondary): Exponents & Exponential Expressions
- MTTC Math (Secondary): Roots & Radical Expressions
- MTTC Math (Secondary): Scientific Notation
- MTTC Math (Secondary): Number Theory
- MTTC Math (Secondary): Ratios & Rates
- MTTC Math (Secondary): Proportions
- MTTC Math (Secondary): Vectors, Matrices & Determinants
- MTTC Math (Secondary): Sequences
- MTTC Math (Secondary): Series
- MTTC Math (Secondary): Factoring
- MTTC Math (Secondary): Properties of Functions
- MTTC Math (Secondary): Graphing Functions
- MTTC Math (Secondary): The Coordinate Graph & Graph Symmetry
- MTTC Math (Secondary): Linear Equations
- MTTC Math (Secondary): Systems of Equations
- MTTC Math (Secondary): Inequalities
- MTTC Math (Secondary): Quadratic Equations
- MTTC Math (Secondary): Working with Quadratic Equations
- MTTC Math (Secondary): Polynomial Functions
- MTTC Math (Secondary): Higher-Degree Polynomial Functions
- MTTC Math (Secondary): Piecewise Functions
- MTTC Math (Secondary): Rational Expressions & Functions
- MTTC Math (Secondary): Exponential & Logarithmic Functions
- MTTC Math (Secondary): Trigonometric Functions
- MTTC Math (Secondary): Solving Trigonometric Equations
- MTTC Math (Secondary): Limits
- MTTC Math (Secondary): Continuity of a Function
- MTTC Math (Secondary): Rate of Change & Derivatives
- MTTC Math (Secondary): Calculating Derivatives & Derivative Rules
- MTTC Math (Secondary): Graphing Derivatives & L'Hopital's Rule
- MTTC Math (Secondary): Applications of Derivatives
- MTTC Math (Secondary): Integration & Integration Techniques
- MTTC Math (Secondary): Integration Applications
- MTTC Math (Secondary): Differential Equations
- MTTC Math (Secondary): Units & Systems of Measurement
- MTTC Math (Secondary): Perimeter & Area
- MTTC Math (Secondary): Polyhedrons & Geometric Solids
- MTTC Math (Secondary): Symmetry, Similarity & Congruence
- MTTC Math (Secondary): Points, Lines & Angles
- MTTC Math (Secondary): Triangles
- MTTC Math (Secondary): Triangle Theorems & Proofs
- MTTC Math (Secondary): Similar Triangles & Polygons
- MTTC Math (Secondary): The Pythagorean Theorem
- MTTC Math (Secondary): Quadrilaterals
- MTTC Math (Secondary): Circles
- MTTC Math (Secondary): Circular Arcs & Measurement
- MTTC Math (Secondary): Conic Sections
- MTTC Math (Secondary): Polar Coordinates & Parametric Equations
- MTTC Math (Secondary): Transformations
- MTTC Math (Secondary): Types of Graphs, Tables, & Data
- MTTC Math (Secondary): Data & Statistics
- MTTC Math (Secondary): Collecting & Analyzing Data
- MTTC Math (Secondary): Samples & Populations in Research
- MTTC Math (Secondary): Probability
- MTTC Math (Secondary): Discrete & Finite Math
- MTTC Mathematics (Secondary) Flashcards