Interactive Course

Integral Calculus

Take the next step on the calculus journey with integrals and sums.

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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.

Topics covered

  • Applications of infinite sums
  • Applications of integrals
  • Convergence tests
  • Integration by parts
  • Partial fractions
  • Polynomial integrals
  • Power series
  • Riemann sums
  • Substitution rule
  • The Fundamental Theorem
  • Transcendental integrals
  • Trigonometric integrals

Interactive quizzes

29

Concepts and exercises

310+

Course map

Prerequisites and Next Steps

  1. 1

    Introduction

    The core of integral calculus.

    1. Calculating Distance

      Find your way to Riemann sums through a simple motion problem.

      1
    2. Different Problem, Same Idea

      See how Riemann sums crop up in many different places.

      2
    3. The Definite Integral

      Leverage your understanding of Riemann sums to master the basic ideas of integration.

      3
  2. 2

    Integration Techniques

    The essential toolkit, from the fundamental theorem to substitutions.

    1. First Calculations

      Make the connection between limits, Riemann sums, and definite integrals.

      4
    2. The Fundamental Theorem of Calculus

      Solidify your complete comprehension of the close connection between derivatives and integrals.

      5
    3. Integrating Polynomials

      Begin to unravel basic integrals with antiderivatives.

      6
    4. Substitution

      Reverse the chain rule to compute challenging integrals.

      7
    5. Exponentials and Trig

      Dabble with transcendental integrals.

      8
  3. 3

    Advanced Integration

    A medley of integration techniques that no professional can do without.

    1. Integration by Parts

      Combine derivative rules to reshape a definite integral into something simpler.

      9
    2. Basics of Partial Fractions

      Learn to break rational integrals down into simpler parts.

      10
    3. Trigonometric Integrals

      Simplify tricky trig integrals with Euler's formula.

      11
    4. Trig Substitution

      Uncover powerful integrals through trigonometry.

      12
    5. Integration in the World

      Look at how integrals are really done in practice.

      13
  4. 4

    Integral Applications

    Use integrals to solve crucial problems in math and science.

    1. Area and Probability

      Dive into some real-world applications of integrals.

      14
    2. Calculating Volume

      Construct volume integrals for a variety of 3D objects using disks, washers, and shells.

      15
    3. Arc Length and Surface Area

      Apply integrals to geometry and explore the paradoxical shape of Gabriel's Horn.

      16
    4. Integration in Physics I

      Go beyond geometry and put integrals to work in physics problems.

      17
    5. Application: Differential Equations

      Apply partial fractions to important real-world problems.

      18
    6. Integration in Physics II

      Bring all integral methods to bear on a crucial physics problem.

      19
  5. 5

    The Idea of Infinite Sums

    The fundamental concepts of sums without end.

    1. Zeno's Paradox & Sums

      Learn about infinite series through a legendary footrace.

      20
    2. Sums and Sigma Notation

      Master the language of sums both finite and infinite.

      21
    3. Secret Identities

      Expose familiar functions as infinite sums in disguise.

      22
  6. 6

    Infinite Series

    Convergence tests for infinite sums and their applications.

    1. Converging Sums

      What does it mean for a sum to converge?

      23
    2. Ratio and Root Tests

      Exploit the geometric sum to create brand new convergence tests.

      24
    3. The Integral Test

      Connect infinite sums to integrals through geometry.

      25
    4. Rearrangements: 0=1?

      Explore the strange properties of sums that don't converge absolutely.

      26
    5. Limit Comparison Test

      Expand the menagerie of convergence tests and create a new fundamental number in the process.

      27
    6. Power Series

      Practice with infinite sums at the heart of many real-world applications.

      28
    7. Basics of Fourier Series

      Step into the world of infinite sums of sines and cosines.

      29