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 ↓


Applications of Integration


Integration is a fundamental concept in calculus, dealing with the accumulation of quantities, such as the area under a curve, volume, and more. Let's look at some of the major applications of integration in simple terms and with visual examples to build a strong foundation.

1. Area under the curve

The area under a curve is a common application of integration. If you have a function y = f(x) and you want to find the area between the curve and the x-axis from x = a to x = b, you use definite integration:

a = ∫[a, b] f(x) dx

You can view it as follows:

y = f(x) A B

Imagine plotting points along a curve from point A to point B and then summing the areas of the thin rectangles drawn under the curve. This idea of "summing thin rectangles" turns into a process of integration.

Example problem: Calculate the area

Consider the function f(x) = x^2. Find the area between f(x) and the x-axis from x = 0 to x = 2.

f(x) = x^2

∫[0, 2] x^2 dx = [x^3/3][0, 2] = (8/3) - (0/3) = 8/3

Thus, the area is 8/3 square units.

2. Volume of solids of revolution

Another application of integration is to find the volume of a solid of revolution. When you rotate a curve around an axis, it creates a three-dimensional shape (such as a vase or donut).

The volume can be calculated using the disk method for rotation around the x-axis:

v = π ∫[a, b] (f(x))^2 dx

Visualizer for understanding:

Revolving Area

The curve is rotated around the x-axis, producing a disc-shaped solid. Integration calculates the total volume by collecting these infinitesimal discs.

Example problem: Find the volume

Consider rotating the curve f(x) = x^2 around the x-axis from x = 0 to x = 1.

V = π ∫[0, 1] (x^2)^2 dx
  = π ∫[0, 1] x^4 dx
  = π[x^5/5][0, 1]
  = π (1/5 - 0/5)
  = π/5

Hence the volume of the solid is π/5 cubic units.

3. Average value of a function

Integration helps to determine the average value of a function over a certain interval. If a function is continuous on [a, b], then the average value f_avg is given by:

f_avg = (1/(ba)) ∫[a, b] f(x) dx

This provides the average height of the function over the interval, which is useful in a variety of practical scenarios.

Example problem: Find the average value

Find the average value of f(x) = 3x^2 on the interval [0, 2].

f_avg = (1/(2-0)) ∫[0, 2] 3x^2 dx
      = (1/2)[3x^3/3][0, 2]
      = (1/2)(24/3 - 0/3)
      = 4

The average value of the function is 4.

4. Work done by the force

In physics, integration can be used to calculate the work done by a force accelerating an object. If a force F(x) accelerates an object along the x-axis from x = a to x = b, then the work done W is given by:

w = ∫[a, b] f(x) dx

This approach is widely applied in physics and engineering, and is useful in calculating the work done by variable forces.

Example problem: Calculate the work done

If a variable force F(x) = 2x moves a particle from x = 1 to x = 3, find the work done.

w = ∫[1, 3] 2x dx
  = [x^2][1, 3]
  = (9 - 1)
  = 8

The work done is 8 units.

5. Fluid pressure and force

Integration helps to determine the pressure force exerted by a fluid on an immersed surface. The force F is calculated using:

,

where ρ is the density of the fluid, g is gravity, h is the height of the fluid, and w(y) is the width at depth y.

Example problem: Find the fluid force

Find the force on a vertical rectangular plate immersed vertically in water (ρ = 1000 kg/m³). The height is 10 m, the width is 2 m, the depth is from 0 to 10 m.

F = ∫[0, 10] (1000 kg/m³)(9.8 m/s²)(10-y)(2) dy
  = 19600 ∫[0, 10] (10-y)dy
  = 19600 [(10y - y²/2)] [0, 10]
  = 19600 (100 - 0 - 50 + 0)
  = 1960000 N

The force applied is 1960000 N

Conclusion

Integration provides a powerful toolset for solving real-world problems. Its applications are vast and varied, from calculating areas to determining volumes, finding averages of functions, and much more. Understanding these fundamental concepts prepares you well for more advanced aspects of calculus and its practical applications.


Grade 11 → 4.5


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Applications of Integration

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