# Ch 30: PLACE Mathematics: Area Under the Curve & Integrals

### About This Chapter

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

As part of your preparation for the PLACE Mathematics test, watch these videos on the use of Riemann sums and calculating integrals. In addition to these topics, our instructors also cover:

- Riemann sums for functions and graphs
- Limits of Riemann sums
- Definite integrals, how to calculate them and their linear properties
- The average value theorem and the fundamental theorem of calculus
- Indefinite integrals

After each lesson video, complete the corresponding quizzes to fortify your knowledge and discover any areas you don't quite understand. Utilize video tags and lesson transcripts in your review of these areas and ask our instructors any questions you may have to effectively master the material before you take the PLACE Mathematics test.

### PLACE Mathematics: Area Under the Curve & Integrals Chapter Objectives

The PLACE Mathematics exam must be completed in four hours and 30 minutes and is a paper-based, multiple-choice test. Of all of the questions on the exam, 19% are part of the calculus and discrete mathematics section. This section is where you could find questions related to the topics in this test. Use these videos and quizzes to prepare for questions on finding the area under a curve and integrals on this test.

### 1. How to Use Riemann Sums for Functions and Graphs

A Riemann sum represents an area that is large and complex as the sum of many smaller, simpler areas. Learn how to use Riemann sums for functions and graphs by using the example of determining the size of a piece of land and using multiple areas.

### 2. How to Find the Limits of Riemann Sums

The area, or Riemann sum, is contained by boundaries, known as limits, that can be calculated. Learn about the concept of Riemann sums, where they are used, and how the limits and integrals can be defined mathematically.

### 3. Definite Integrals: Definition

A definite integral is found as the limit between a line graphed from an equation, and the x-axis, either positive or negative. Learn how this limit is identified in practical examples of definite integrals.

### 4. How to Use Riemann Sums to Calculate Integrals

Riemann sums use the method of 'slicing' the area of a graph to isolate the equation used to calculate definite integrals. Follow example problems of using Riemann sums to find an area even when divided into different sections.

### 5. Linear Properties of Definite Integrals

The linear properties of definite integrals allow complex problems to be solved. Learn how to differentiate between and to use the zero integral property, backward property, constant property, additive property, and sums property.

### 6. Average Value Theorem

The calculation of the average values of equations using integrals can be done using the average value theorem. Learn more about the average value theorem, how to display it on graphs, and how to calculate continuous functions.

### 7. The Fundamental Theorem of Calculus

The fundamental theorem of calculus is one of the most important points to understand in mathematics. Learn to define the formula of the fundamental theorem of calculus and explore examples of it put into practice.

### 8. Indefinite Integrals as Anti Derivatives

Indefinite integrals, an integral of the integrand which does not have upper or lower limits, can be used to identify individual points at specific times. Learn more about the fundamental theorem, use of antiderivatives, and indefinite integrals through examples in this lesson.

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### Other Chapters

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

- PLACE Mathematics: Properties of Real Numbers
- PLACE Mathematics: Fractions
- PLACE Mathematics: Decimals & Percents
- PLACE Mathematics: Ratios & Proportions
- PLACE Mathematics: Measurements & Conversions
- PLACE Mathematics: Logic
- PLACE Mathematics: Mathematical Reasoning
- PLACE Mathematics: Vector Operations
- PLACE Mathematics: Matrices & Determinants
- PLACE Mathematics: Exponents & Exponential Expressions
- PLACE Mathematics: Algebraic Expressions
- PLACE Mathematics: Linear Equations
- PLACE Mathematics: Inequalities
- PLACE Mathematics: Absolute Value Problems
- PLACE Mathematics: Quadratic Equations
- PLACE Mathematics: Polynomials
- PLACE Mathematics: Rational Expressions
- PLACE Mathematics: Radical Expressions
- PLACE Mathematics: Systems of Equations
- PLACE Mathematics: Complex Numbers
- PLACE Mathematics: Functions
- PLACE Mathematics: Graphing Piecewise Functions
- PLACE Mathematics: Exponential and Logarithmic Functions
- PLACE Mathematics: Continuity of Functions
- PLACE Mathematics: Limits
- PLACE Mathematics: Rate of Change
- PLACE Mathematics: Calculating Derivatives & Derivative Rules
- PLACE Mathematics: Graphing Derivatives & L'Hopital's Rule
- PLACE Mathematics: Applications of Derivatives
- PLACE Mathematics: Integration & Integration Techniques
- PLACE Mathematics: Integration Applications
- PLACE Mathematics: Foundations of Geometry
- PLACE Mathematics: Introduction to Geometric Figures
- PLACE Mathematics: Properties of Triangles
- PLACE Mathematics: Triangles, Theorems & Proofs
- PLACE Mathematics: Parallel Lines & Polygons
- PLACE Mathematics: Quadrilaterals
- PLACE Mathematics: Circular Arcs & Circles
- PLACE Mathematics: Conic Sections
- PLACE Mathematics: Geometric Solids
- PLACE Mathematics: Analytical Geometry
- PLACE Mathematics: Using Trigonometric Functions
- PLACE Mathematics: Trigonometric Graphs
- PLACE Mathematics: Solving Trigonometric Equations
- PLACE Mathematics: Trigonometric Identities
- PLACE Mathematics: Sequences & Series
- PLACE Mathematics: Graph Theory
- PLACE Mathematics: Set Theory
- PLACE Mathematics: Overview of Statistics
- PLACE Mathematics: Summarizing Data
- PLACE Mathematics: Tables & Plots
- PLACE Mathematics: Probability
- PLACE Mathematics: Discrete Probability Distributions
- PLACE Mathematics: Continuous Probability Distributions
- PLACE Mathematics: Sampling
- PLACE Mathematics: Hypothesis Testing
- PLACE Mathematics: Regression & Correlation
- PLACE Mathematics Flashcards