About This Chapter
Integration - Chapter Summary
In this engaging chapter, you'll learn all about integration in mathematics, including the definition of definite integrals, using anti-derivatives to calculate the indefinite integrals of polynomials and the trapezoid rule. Our short, user-friendly lessons also cover the formula associated with Simpson's Rule and the use of substitution to solve integrals. Other topics include integration by parts and finding simple areas with integration and root finding. Once you complete this chapter, you should be ready to:
- Integrate functions with partial fractions
- Give an example of partial fraction decomposition
- Complete the square to integrate functions
- Use integration to find volumes of revolution
- Solve practice problems involving the volume of a solid of revolution
You can work at your own pace as you access these learning tools on your mobile phone, tablet or computer. All of our lessons are followed by a multiple-choice quiz so you can test your understanding of the subjects. Feel free to watch the video lessons as many times as you need to in order to master the information, or use the video tabs feature to skip directly to a specific section.
1. 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.
2. Anti-Derivatives: Calculating Indefinite Integrals of Polynomials
If you throw a ball in the air, the path that it takes is a polynomial. In this lesson, learn how to integrate these fantastic functions by putting together your knowledge of the fundamental theorem of calculus and your ability to differentiate polynomial functions.
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. What is Simpson's Rule? - Example & Formula
In this lesson, you'll learn how to approximate the integration of a function using a numerical method called Simpson's Rule. This method is particularly useful when integration is difficult or even impossible to do using standard techniques.
5. How to Solve Integrals Using Substitution
Some integrals are as easy as riding a bike. But more often, integrals can look like deformed bikes from Mars in the year 3000. In this lesson, you will learn how to transform these scary-looking integrals into simpler ones that really are as easy as riding a bike.
6. Using Integration By Parts
Your mother may have warned you not to bite off more than you can chew. The same thing is true with integration. In this lesson, learn how integration by parts can help you split a big interval into bite-sized pieces!
7. How to Find Simple Areas With Root Finding and Integration
Combine your calculus tools in this lesson! Find the area enclosed by multiple arbitrary functions by first finding the root of an equation and then integrating over the resulting range.
8. How to Integrate Functions With Partial Fractions
In this lesson, learn how to integrate complicated fractions by using the partial fractions technique. That is, you will turn a complicated fraction into something a bit easier to integrate by finding partial fractions!
9. Partial Fraction Decomposition: Rules & Examples
What if you had a way to expand certain large math expressions into smaller pieces? This would make some calculus integrals easier to solve. In this lesson, we explore such a method: partial fraction decomposition.
10. How to Integrate Functions by Completing the Square
We want to integrate a rational function that includes a polynomial by expressing the polynomial as the sum or difference of two squares so that we can apply a substitution (trigonometric or logarithmic).
11. How to Find Volumes of Revolution With Integration
Some shapes look the same as you rotate them, like the body of a football. In this lesson, learn how to find the volumes of shapes that have symmetry around an axis using the volume of revolution integration technique.
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Other chapters within the HSC Mathematics: Exam Prep & Syllabus course
- Mathematical Proofs & Reasoning
- Basic Arithmetic & Algebra
- Factorization of Binomials & Trinomials
- Types of Equations & Inequalities
- Plane Geometry
- Circle Plane Geometry
- Probability & Randomization
- Arithmetic & Geometric Series
- Trigonometric Ratios
- Geometrical Representations of Functions
- Linear Functions & Lines
- Derivatives of Functions
- Quadratic Polynomials & Parabolas
- Geometrical Applications of Differentiation
- Logarithmic & Exponential Functions
- Calculating Trigonometric Functions
- Understanding Inverse Functions
- Factoring & Graphing Polynomials
- Roots & Coefficients of Polynomials
- Binomial Theorem & Probability
- Graphing & Solving Functions
- Solving & Graphing Complex Numbers
- Geometric Representations of Complex Numbers
- Square Roots, Powers & Roots of Complex Numbers
- Conic Sections Basics
- The Ellipse in Algebra
- The Hyperbola
- The Rectangular Hyperbola
- Calculus Applications: Rate of Change
- Calculus Applications: Velocity & Acceleration
- Calculus Applications: Projectile & Harmonic Motion
- Calculus Applications: Resisted Motion
- Calculus Applications: Circular Motion
- HSC Mathematics Flashcards