Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus → Integral Calculus ↓


Definite Integrals


The definite integral is a fundamental concept in calculus, used to calculate accumulations of quantities such as area, volume, work, and other physical properties. Unlike the indefinite integral, which represents a family of functions, a definite integral returns a single number. It essentially calculates the net area under a curve over an interval, providing a very powerful tool for analysis.

    ∫ a b f(x) dx

Basic concepts of definite integrals

When we talk about definite integrals, we are interested in finding the area under a curve described by a function between two points on the x-axis, say a and b.

The notation for the definite integral of a function f(x) from a to b is:

    ∫ a b f(x) dx

You can visualize that gap like this: A B

Understanding integrals

To calculate a definite integral, we essentially divide the area under the curve into thin vertical strips and sum their areas. Each strip is approximately a rectangle and the more strips we use, the better our estimate will be. The thickness of each strip approaches zero over the range of our calculations.

Consider the following function, for simplicity let's take f(x) = x^2. We are interested in the interval from a = 0 to b = 2. 

    f(x) = x^2
The integral setup is: 
    ∫ 0 2 x^2 dx

Example calculation

Let's calculate the definite integral of f(x) = x^2 from 0 to 2 step by step:

Step 1: Find the antiderivative of f(x)

The antiderivative of f(x) = x^2 is found by raising the power by one and dividing by the new power. 

        f(x) = (1/3)x^3 + c

Step 2: Evaluate the antiderivative from a to b

Using the fundamental theorem of calculus, we evaluate the antiderivative at the boundaries of the interval and reduce: 

        f(b) - f(a) = [(1/3)(2)^3 + c] - [(1/3)(0)^3 + c]
                    = [(1/3)(8)] - [0]
                    = 8/3

Fundamental theorem of calculus

The fundamental theorem of calculus connects the concept of differentiation to integration. It states that if F is the antiderivative of f on the interval [a, b], then 

        ∫ a b f(x) dx = f(b) - f(a)

This theorem not only provides a way to compute a definite integral by evaluating the antiderivative, but also establishes a deep connection between the geometry of a function (represented by the integral) and its calculus description.

Geometrical interpretation

Suppose the function f(x) is a simple curve on the xy-plane, and we are finding the area under this curve from point a to point b.

In this illustration, the blue shaded area represents the area under the curve: A B

Working with practical examples

Example 1: Finding the area under a curve

Suppose you have a function f(x) = 3x^2 and you want to find the area under the curve from x = 1 to x = 4.

Step 1: Find the antiderivative of f(x) 

         f(x) = ∫ 3x^2 dx = x^3 + c

Step 2: Evaluate the antiderivative from 1 to 4 

        f(4) - f(1) = [(4)^3] - [(1)^3]
                    = [64] - [1]
                    = 63

Example 2: Total distance travelled

If a chemical solution is being pumped into a tank at the rate of R(t) = 5t - 2 liters per hour, how much solution is pumped into the tank from t = 1 to t = 5 hours?

Step 1: Find the antiderivative of R(t) 

        ∫ (5t - 2)dt = (5/2)t^2 - 2t + c

Step 2: Rate from 1 to 5 

        f(5) - f(1) = [(5/2)(5)^2 - 2(5)] - [(5/2)(1)^2 - 2(1)]
                    = [(5/2)(25) – 10] – [(5/2) – 2]
                    = [62.5 – 10] – [2.5 – 2]
                    = 52.5 - 0.5
                    = 52

Pythagorean tuning: Frequency ratios

Even though this may extend beyond traditional calculus, consider how frequency intervals are related through ratios. If you work with a frequency ratio function with intervals:

    r(f) = 440 * 2^(x/12)
where x is the number of semitones from 440 Hz. To find the accumulated "increase" in frequency from x = 0 to x = 12 (one octave), calculate:

Step 1: Integral setup 

    ∫ 0 12 440 * 2^(x/12) dx

Step 2: Solve the integral

Applying the Fundamental Theorem and Rules of Calculus: 

Let y = 2^(x/12) \
Then, we differentiate d/dx[y] = ln(2)/12 * 2^(x/12) \
∫ 440 * y dy = 440 * 12/ln(2) * [y^2/2 - y/2] = 5280/ln(2) * [2 - 1]

The solution describes the evolution under a musical interval, providing fascinating insight!

Conclusion

Definite integrals are an essential component of calculus that provide us with tools to calculate areas under curves, total changes in quantities, and a vast number of other applications that have profound implications in both theoretical mathematics and practical scenarios in a variety of fields including physics, engineering, economics, and beyond. They connect us deeply to geometric intuition and algebraic computation.

As you further explore this rich field, you will dive deeper into various techniques such as numerical integration, improper integration, and applications in differential equations, as well as more advanced integration techniques and real-world applications, enriching your understanding and mathematical toolkit for solving complex problems.


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