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 ↓


Area Under a Curve


Calculus is an essential branch of mathematics, and one of its primary components is the concept of integration, which helps us find the area under a curve. This idea is fundamental in mathematics, physics, engineering, and many other fields. In simple terms, integration can be seen as the opposite process of differentiation, which is another important concept in calculus. In this detailed explanation, we will explore the idea of the area under a curve, understand it using several text examples, and visualize it with illustrations.

What is the area under the curve?

When we talk about the area under a curve in mathematics, particularly calculus, we refer to the total space enclosed between a given curve, the x-axis, and specified limits on the x-axis. More formally, if we have a function y = f(x) that is continuous on an interval [a, b], the area under the curve from x = a to x = b can be calculated using integration.

The mathematical expression for this can be represented as:

∫ f(x) dx from a to b

This expression is read as "the integral of f(x) from a to b." Here, dx indicates that we are taking an integral with respect to x.

Understanding integration

Integration is similar to a summation process where the total area is found by adding up the infinitesimally small areas under the curve. This is similar to decomposing shapes into smaller parts and finding their area. The basic idea of integration is to break down complex curves into infinitesimally small rectangles and then add up their areas to get the total area under the curve.

Example of rectangular approximation

Consider a simple curve such as y = x^2 from x = 0 to x = 2 To find the area under this curve using the approximation method, we can divide this interval into smaller subintervals, calculate the area of the rectangles, and sum them. The smaller the rectangles, the more accurate our approximation will be.

The image above shows the curve of y = x^2 and the rectangles underneath it. The height of each rectangle represents the value of the function over a particular subinterval. By increasing the number of rectangles, we estimate the area under the curve more accurately.

Definite and indefinite integrals

In calculus, when we calculate the area under a curve, we often refer to definite and indefinite integrals. Understanding the difference between these two can help clarify the role of integration in finding areas.

Indefinite integral

The indefinite integral, also called the antiderivative, is a general function that is obtained by reversing the differentiation process. The indefinite integral of the function f(x) is represented as:

∫ f(x) dx = f(x) + c

Where F(x) is the antiderivative and C is the constant of integration. The indefinite integral represents a family of functions.

Definite integral

A definite integral has limits, which means that it integrates a function over a specified interval. This provides a number that shows the net area between the curve and the x-axis. The basic theorem of calculus connects indefinite and definite integrals, showing that if F(x) is an antiderivative of f(x), then:

∫ f(x) from a to b dx = F(b) - F(a)

This equation states that the area from x = a to x = b is the difference between the values of the antiderivative F(x) at b and a.

Visual exploration of integration

Let's illustrate this concept visually with another example. Suppose we have the function y = 3x^2 and we want to find the area under this curve from x = 1 to x = 3.

1 3

The curve in the view represents y = 3x^2 and the shaded area indicates the area we want to find. By evaluating the definite integral:

∫ 3x^2 dx from 1 to 3

We start by finding the antiderivative of 3x^2, which is x^3. So, the area under the curve from 1 to 3 is:

[x^3] from 1 to 3 = 3^3 - 1^3 = 27 - 1 = 26

Therefore, the area under y = 3x^2 from x = 1 to x = 3 is 26 square units.

Practical lesson example

Let's further strengthen our understanding with additional practical lesson examples. Consider calculating the area under the curve for the function y = 5x from x = 2 to x = 5.

The integral becomes:

∫ 2 to the 5 of 5x dx

Finding the antiderivative, we get:

(5/2)x^2 + c

Therefore, the area is:

[(5/2)x^2] from 2 to 5 = (5/2)(5^2) - (5/2)(2^2) = (5/2)(25) - (5/2)(4) = 62.5 - 10 = 52.5

Thus, the area under y = 5x from 2 to 5 is 52.5 square units.

When the curve lies below the X-axis

When a curve is below the x-axis, the definite integral gives a negative value, indicating that the curve is below the x-axis. However, when calculating the actual area, this negative sign is usually ignored, since the area is naturally non-negative.

For example, if we have y = -2x and we want to calculate the area from x = 0 to x = 3, we evaluate:

∫ -2x dx from 0 to 3

Due to this:

(-2/2)x^2 from 0 to 3 = -3^2 + 0^2 = -9

Even though the result is -9, the area is considered to be only 9 square units.

Applications in the real world

Understanding the area under a curve is not just a mathematical exercise; it has meaningful applications in the real world. Here are some examples:

  • Physics: The area under an object's velocity-time graph represents its displacement. By integrating the velocity function over time, you can determine the total change in position.
  • Economics: In economics, integrating a supply curve or demand curve over an interval can provide information about consumer or producer surplus.
  • Biology: Integration helps calculate population growth when the growth rate over time is known as a function.
  • Engineering: In civil engineering, integrating stress–strain curves is valuable for predicting the behavior of materials under various loads.

Conclusion

The area under the curve is a powerful concept that brings together integration and real-world problem-solving. Whether estimating physical quantities, understanding economic models, or solving engineering problems, mastering this concept enhances your mathematical skills. Practice and understanding on these basic examples can make dealing with complex curves more manageable and practical.


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Area Under a Curve

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