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Brilliant

Integral Calculus

via Brilliant

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Overview

In a sense, differential calculus is local: it focuses on aspects of a function near a given point, like its rate of change there. Integral calculus complements this by taking a more complete view of a function throughout part or all of its domain.

This course provides complete coverage of the two essential pillars of integral calculus: integrals and infinite series. By the end, you'll know their core principles and how to apply them to problems in geometry, probability, and physics.

Syllabus

  • Introduction: The core of integral calculus.
    • Calculating Distance: Find your way to Riemann sums through a simple motion problem.
    • Using Riemann Sums: See how Riemann sums crop up in many different places.
    • The Definite Integral: Leverage your understanding of Riemann sums to master the basic ideas of integration.
  • Integration Techniques: The essential toolkit, from the fundamental theorem to substitutions.
    • First Calculations: Make the connection between limits, Riemann sums, and definite integrals.
    • The Fundamental Theorem of Calculus: Solidify your complete comprehension of the close connection between derivatives and integrals.
    • Integrating Polynomials: Begin to unravel basic integrals with antiderivatives.
    • Substitution: Reverse the chain rule to compute challenging integrals.
    • Exponentials and Trig: Dabble with transcendental integrals.
  • Advanced Integration: A medley of integration techniques that no professional can do without.
    • Integration by Parts: Combine derivative rules to reshape a definite integral into something simpler.
    • Basics of Partial Fractions: Learn to break rational integrals down into simpler parts.
    • Trigonometric Integrals: Simplify tricky trig integrals with Euler's formula.
    • Trig Substitution: Uncover powerful integrals through trigonometry.
    • Integration in the World: Look at how integrals are really done in practice.
  • Integral Applications: Use integrals to solve crucial problems in math and science.
    • Area and Probability: Dive into some real-world applications of integrals.
    • Calculating Volume: Construct volume integrals for a variety of 3D objects using disks, washers, and shells.
    • Arc Length and Surface Area: Apply integrals to geometry and explore the paradoxical shape of Gabriel's Horn.
    • Integration in Physics I: Go beyond geometry and put integrals to work in physics problems.
    • Application: Differential Equations: Apply partial fractions to important real-world problems.
    • Integration in Physics II: Bring all integral methods to bear on a crucial physics problem.
  • The Idea of Infinite Sums: The fundamental concepts of sums without end.
    • Zeno's Paradox & Sums: Learn about infinite series through a legendary footrace.
    • Sums and Sigma Notation: Master the language of sums both finite and infinite.
    • Secret Identities: Expose familiar functions as infinite sums in disguise.
  • Infinite Series: Convergence tests for infinite sums and their applications.
    • Converging Sums: What does it mean for a sum to converge?
    • Ratio and Root Tests: Exploit the geometric sum to create brand new convergence tests.
    • The Integral Test: Connect infinite sums to integrals through geometry.
    • Rearrangements: 0=1?: Explore the strange properties of sums that don't converge absolutely.
    • Limit Comparison Test: Expand the menagerie of convergence tests and create a new fundamental number in the process.
    • Power Series: Practice with infinite sums at the heart of many real-world applications.
    • Basics of Fourier Series: Step into the world of infinite sums of sines and cosines.

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