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 ↓


Definite Integrals


Welcome to the discovery of the definite integral, an important concept in calculus. Definite integrals are used to find the total quantity accumulated, such as the area under a curve, the volume of a solid, or other quantities when functions are involved.

Understanding the basics

Integration is a fundamental concept of calculus that combines two different ideas: cumulation and area. Initially, we encounter differentiation, where we find rates of change. Integration is, in a sense, the opposite process. There are two primary types of integrals in calculus:

  • Indefinite integrals, which represent families of antiderivative functions.
  • Definite integrals, which provide a number representing the accumulated change over an interval.

When you perform a definite integral, you are essentially calculating the "total accumulation" of a function over a specific interval. The result of a definite integral is a number, not a function.

Integral symbol

The symbol for unification is an elongated S: ∫, for "sum," reflecting its roots in addition and cumulation.

Definite integral notation

The definite integral of a function f(x) from a to b is represented as:

∫ a b f(x) dx

Here:

  • a and b are the lower and upper limits of the integral, respectively.
  • f(x) is the function that is being integrated.
  • dx indicates that the integration is with respect to the variable x.

The value of the definite integral from x = a to x = b represents the net area between the function f(x) and x axis.

Geometrical interpretation

Geometrically, the definite integral of a function corresponds to the areas under the curve of the function over some interval on the x-axis. The definite integral is the net area, with areas above the x-axis considered positive and areas below negative.

Let us consider a simple example:

A B f(x)

The above diagram shows the curve of a continuous function f(x) from a to b. The shaded region represents the definite integral of f(x) from a to b.

Evaluating definite integrals

The definite integral is calculated using the fundamental theorem of calculus. This theorem connects differentiation and integration, showing that they are, in essence, inverse processes.

Fundamental theorem of calculus

This theorem can be stated as follows:

If F(x) is the antiderivative of f(x) on the interval [a, b], then
∫ a b f(x) dx = F(b) - F(a)

This tells us that we can find the definite integral of a function if we can find its antiderivative.

Example calculation

Let's calculate a simple definite integral: Find the definite integral of f(x) = x^2 from x = 1 to x = 3.

First, find the antiderivative of f(x) = x^2, which we denote by F(x).

Recall that the antiderivative of x^n is (1/(n+1))x^(n+1) + C.

The antiderivative of x^2 is:

F(x) = (1/3)x^3

Now, apply the fundamental theorem of calculus:

∫ 1 3 x^2 dx = F(3) - F(1) = (1/3)(3)^3 - (1/3)(1)^3

Calculate the value:

F(3) = (1/3)(27) = 9
F(1) = (1/3)(1) = 1/3

The definite integral is:

9 - 1/3 = 8.67

Thus, the area under the curve f(x) = x^2 from x=1 to x=3 is approximately 8.67.

Properties of definite integrals

Like arithmetic operations, definite integrals have a number of useful properties:

1. Additivity

If c is a point between a and b, then:

∫ a b f(x) dx = ∫ a c f(x) dx + ∫ c b f(x) dx

This property allows us to break down an integral into smaller, more manageable parts.

2. Linearity

Definite integrals respect the distributive property:

∫ a b [cf(x) + dg(x)] dx = c∫ a b f(x) dx + d∫ a b g(x) dx

where c and d are constants. This allows us to break down integrals involving sums or constant multiples.

3. Reversal of boundaries

If you change the limits of the integral, the sign of the integral changes:

∫ a b f(x) dx = -∫ b a f(x) dx

This property is helpful when you accidentally reverse your limits of integration.

4. Integration of zero

The integral of zero is always zero:

∫ a b 0 dx = 0

This reflects the idea that if there is no change, then the accumulated change should also be zero.

Practical applications of definite integrals

Definite integrals have many practical applications in various fields:

  • Physics: Calculating work done by a force, area under velocity-time graph to find displacement.
  • Economics: Finding consumer and producer surplus.
  • Engineering: Designing systems that rely on understanding stored quantities or limitations, such as heat generation.
  • Biology: Modelling population change and understanding growth rates over time.

Conclusion

The concept of the definite integral is not only a deep principle within mathematics, but also a valuable tool used across science and engineering disciplines. By understanding how to work with definite integrals, you will be able to solve practical problems involving cumulative processes or finding values belonging to specific intervals.

Through the lens of definite integrals, many real-world applications become understandable, making it an essential topic to fully understand.


Grade 11 → 4.4.3


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