Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Calculus ↓


Integration


Integration is a fundamental concept in calculus that is about adding parts to find the whole. Imagine you have a pile of small pieces. Each piece is so small that individually, they don't make much of an impact. However, when you collect these pieces and put them together, they can form something significant. This is what integration tries to do with mathematical functions.

The idea behind the integration

Integration is often explained using the example of summing areas. When you integrate a function, you are essentially calculating the area under its graph. This 'graph' is often a curve, and it can be difficult to find the area directly. Integration is a tool that helps make this process easier.

X-axis Shaft f(x) Area

Key Terms in Integration

Before we dive deeper into how the integration works, let's look at some key terms:

  • Integral: The result of an integration operation.
  • Definite Integral: Represents the accumulation of quantities, such as areas, between a start and end point.
  • Indefinite integral: denotes the general form of the antiderivative of a function.
  • Antiderivative: A function whose derivative gives the given function.

Understanding Indefinite Integrals

Indefinite integrals are like working backwards: if you know the rate of change of something (for example, velocity), you can determine the original state (distance traveled).

The general notation for the indefinite integral is:

∫ f(x) dx = F(x) + C

Here, f(x) is your function, F(x) is the antiderivative, and C is the constant of integration.

Example of Indefinite Integration

Let's find the indefinite integral of the function f(x) = 2x.

This process involves finding a function F(x) such that:

F'(x) = 2x

In this case, F(x) = x^2 because

F'(x) = d(x^2)/dx = 2x

Thus, the indefinite integral is:

∫ 2x dx = x^2 + C

Understanding Definite Integrals

On the other hand, the definite integral allows you to calculate the area under a curve between two points (let's call them a and b).

The general notation for the definite integral is:

∫ab f(x) dx

It is calculated using the fundamental theorem of calculus:

∫ab f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

Example of Definite Integration

Let's consider the same function f(x) = 2x, and find the area under the curve from x = 1 to x = 3.

First, as before, find the antiderivative F(x) = x^2.

Using the fundamental theorem of calculus:

∫13 2x dx = [x^2]13 = 3^2 - 1^2 = 9 - 1 = 8

Thus, the area under f(x) = 2x from 1 to 3 is 8 square units.

Relationship Between Integration and Differentiation

Integration and differentiation are opposite processes. While differentiation focuses on rates and slopes of change, integration focuses on restoring the original function from those rates of change (essentially accumulating those changes).

Viewing integration as an accumulation of fields

Think of integration as a way of collecting small pieces of area to find the total area. Let's imagine small rectangular strips under a curve, each extending perpendicularly from the x-axis to the curve. These rectangles estimate the area under the curve.

X-axis Shaft f(x)

As you add more rectangles, making them thinner, the estimate gets better. In the limit, as the width of the rectangles approaches zero, the sum of the areas of the rectangles approaches the actual area under the curve.

General rules of integration

When performing integration, there are several rules that simplify the process, just like you have rules for differentiation. Some of the essential rules are as follows:

Power law

This rule states:

∫ x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1

For n = -1 the integral becomes:

∫ x^(-1) dx = ln|x| + C

Continuation rule

This rule is straightforward and states:

∫ a dx = ax + C

Here, a is a constant. This means that when you integrate a constant, you just have to multiply it by the variable.

Sum rules

This rule says that you can divide integrals by sums:

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

Integration by replacement

This technique is used when substitution can simplify the integration process. It is similar to the chain rule in differentiation.

Given:

∫ f(g(x)) g'(x) dx

Using substitution, let u = g(x) and thus du = g'(x) dx.

The integral transforms as follows:

∫ f(u) du

Example of Integration by Substitution

Integrate ∫ 2x (x^2 + 1)^3 dx.

Choose u = x^2 + 1, then du = 2x dx.

The integral becomes:

∫ u^3 du

Use of power rule:

(u^4)/4 + C

Substitute back:

(x^2 + 1)^4/4 + C

Importance of Constant of Integration

The constant C in the indefinite integral reflects all possible solutions because any constant vanishes when differentiated. For example, x^2 + 3 and x^2 - 7 both have the same derivative, 2x. Without C, the integral would not reflect this family of solutions.

Important functions and their integrals

It is useful to know integrals of normal forms:

  • ∫ e^x dx = e^x + C
  • ∫ sin(x) dx = -cos(x) + C
  • ∫ cos(x) dx = sin(x) + C
  • ∫ 1/x dx = ln|x| + C

Applications of Integration

Finding the Area

As discussed, you can calculate the area under a curve by integrating the function. This is especially useful in physics for finding real-world quantities such as area, volume, and averages.

Physics Applications

In physics, integration is used to find quantities, such as displacement, when you know the velocity function, or work, when you know the force function over a distance.

Volumes of solids of revolution

One can also find the volume of a solid modeled by rotating a function around a given axis, using techniques such as the disk or washer method by integration.

Challenges and suggestions

Integration can be challenging, as sometimes the function to integrate is not easy. Learning to recognize patterns and practicing frequently will improve your skills. It is important to understand the fundamental rules and basic antiderivatives to tackle more advanced topics.

Summary

Integration is a powerful mathematical concept that helps find convex areas, solve differential equations, and much more. Although it may seem challenging at first, understanding the basic concepts and practicing with different functions will make it more intuitive. Keep exploring and applying integration techniques to see how they work in real-world scenarios.


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