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


Integral calculus is one of the two central areas of calculus, the other being differential calculus. While differential calculus focuses on the concept of derivatives and rates of change, integral calculus deals with the concept of integrals and accumulation of quantities. This area of mathematics gives us powerful tools for analyzing areas, volumes, and many other concepts wrapped around accumulation.

What is integration?

Integration is often described as the opposite process of differentiation. While differentiation breaks down a function to study its rates of change, integration creates a cumulative solution by aggregating data. Integrating a function essentially sums an infinite number of infinitesimal quantities. This can be seen by imagining challenges such as adding an infinite number of thin rectangles under a curve, giving us the total area.

Definite integral

The definite integral is represented with the integral sign (∫), a function, a difference of the variable (such as dx), and the limits of integration determined by the lower and upper limits. It shows the area under the function curve from the lower limit to the upper limit. Mathematically, it is expressed as:

∫[a,b] f(x) dx

Here, a and b are the limits of integration, indicating where to start and end the calculation on the x-axis.

Visual example of a definite integral

To understand how the definite integral works, consider the curve of a simple function f(x) = x^2. The definite integral from x = 0 to x = 2 will calculate the area under this curve as follows:

X Y 0 1 2

The shaded region represents the definite integral of the function f(x) = x^2 from the lower limit x = 0 to the upper limit x = 2 Calculating this integral gives the total "area" collected under the curve within those limits.

Indefinite integral

The indefinite integral, unlike the definite integral, does not have limits of integration. Instead, it expresses a family of functions, which are antiderivatives of the original function. It is expressed as:

∫ f(x) dx = F(x) + C

In this context, F(x) is any function whose derivative is f(x), and C denotes an arbitrary constant, indicating a family of parallel functions that differ by a vertical shift.

Fundamental theorem of calculus

The fundamental theorem of calculus connects differentiation and integration, showing that they are essentially inverse processes. It consists of two main parts. The first part states that if F is the antiderivative of f(x) on some interval, then:

∫[a,b] f(x) dx = F(b) - F(a)

The second part of the theorem asserts that if f is continuous on an interval and F is the integral of f, then f is the derivative of F.

Applications of integral calculus

Integral calculus is used in many diverse fields such as physics, engineering, economics, and statistics. Below are some scenarios illustrating its applications:

1. Calculation of the area under the curve

Finding the area under a curve is one of the most crucial objectives of integral calculus. By computing the definite integral of a function, one can efficiently determine the net area bounded by the function and the x-axis.

2. Determining the volume of solid objects

Integral calculus helps solve problems related to volume by rotating curves around an axis. It is most important in engineering fields for evaluating the volume of irregularly shaped objects.

3. Solving differential equations

Integration plays an important role in solving differential equations, and aids in modeling systems in natural and artificial contexts, such as population growth or electrical circuits.

Example - Calculating a simple integral

Let's calculate the simple integral of the function f(x) = 3x^2. Its indefinite integral is:

∫ 3x^2 dx = x^3 + C

This integral represents the family of functions whose derivative is 3x^2.

Integration techniques

1. Substitution method

The substitution method is similar to the reverse of the chain rule in differentiation. Substitution is chosen to simplify the integral, turning a complex expression into a more manageable one.

2. Integration by parts

Integration by parts is applicable when dealing with products of functions. It is derived from the product rule for differentiation and is represented as:

∫ u dv = uv - ∫ v du

Here, u and dv are chosen parts of the integral to facilitate easier integration.

3. Partial fraction decomposition

Partial fraction decomposition breaks rational functions into simpler fractions, making them easier to integrate individually.

Conclusion

Integral calculus emerges as an important branch of mathematics, working in tandem with differential calculus to tackle real-world challenges. Through both definite and indefinite integrals, along with myriad techniques such as substitution and integration by parts, integration provides efficient means to calculate areas, volumes, and other cumulative properties. Indeed, from engineering feats to natural phenomena, integral calculus stands as a vital tool in the mathematical toolkit.


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