Comprehensive Curriculum

Classical topics like Riemann sums, u-substitution, integration by parts, and more

Python Integration

Visualize and implement concepts using Python libraries such as NumPy, SymPy, and Matplotlib

Real-World Applications

Explore applications like computing arc lengths, surface areas, and probability distributions

About the Course

If Calculus 1 was about understanding change, then Calculus 2 is about accumulation: how small changes stack up to build area, volume, probability, and complexity. Integration is where mathematics meets imagination — it's where the abstract and the physical merge. From Riemann sums to probability distributions, from arc lengths to solids of revolution, integral calculus provides the tools to describe, quantify, and visualize everything from the motion of particles to the structure of data. It’s a gateway to multivariable calculus, mathematical modeling, and data science. And it's not just a theoretical subject. Integration is foundational to fields including physics, engineering, machine learning, quantitative finance, and statistics. If you want to understand the algorithms behind data science or build the mathematical foundation needed for AI, you need to understand integrals. Well, you could learn integration by watching a lecture filled with blackboard equations and hoping it sinks in. Or you could take a more interactive, hands-on approach. This course follows the principle that “you can learn a lot of math with a bit of coding.” You'll use Python — especially NumPy, SymPy, and Matplotlib — to visualize integrals, implement numerical approximations, explore convergence, and gain intuition for the fundamental ideas of calculus.

Meet Your Instructor

I am a full-time educator and writer, and former professor of neuroscience. I "retired" from my tenured job so I could focus my time and energy creating high-quality educational material just for you.  I have 20 years of experience teaching coding, data analysis, signal processing, statistics, and math. I've taught undergraduate students, PhD candidates, postdoctoral researchers, and full professors. I have taught in "traditional" university courses, special week-long intensive courses, and Nobel prize-winning research labs. So trust me when I say that that you've come to the right place to maximize your learning from an instructor who has spent two decades refining and perfecting his teaching style.   Over 300,000 students have watched over 50,000,000 minutes of my courses. Come find out why!

Curriculum

14 Chapters · 94 Lessons

  1. 1

    Introductions and course materials

    1. Downloading and using the code Free preview
    2. Prerequisites and how to rock this course Free preview
    3. About the "CodeChallenge" videos Free preview
    4. (Included in full purchase)

      My policy on sharing code

    5. (Included in full purchase)

      Use ChatGPT to learn calculus

  2. 2

    Intuition for integration

    1. (Included in full purchase)

      Section summary and goals

    2. (Included in full purchase)

      Integration as "inverse differentiation"

    3. (Included in full purchase)

      CodeChallenge: Drawing discrete integrals

    4. Integration as geometric area Free preview
    5. (Included in full purchase)

      CodeChallenge: Geometric approximations

    6. (Included in full purchase)

      The dual perspectives of integration

    7. (Included in full purchase)

      The fundamental theorem of calculus, Part 1

    8. (Included in full purchase)

      The fundamental theorem of calculus, Part 2

    9. (Included in full purchase)

      CodeChallenge: Approximate exact integrals 1/2

    10. (Included in full purchase)

      CodeChallenge: Approximate exact integrals 2/2

    11. (Included in full purchase)

      Why integration is harder than differentiation

    12. Calculart! Fun and beauty with calculus :) Free preview
  3. 3

    Tangent: Does math mean anything?

    1. (Included in full purchase)

      How much of math reflects reality?

  4. 4

    Geometric approximations

    1. Section summary and goals Free preview
    2. (Included in full purchase)

      Riemann sums

    3. CodeChallenge: Riemann approximations 1/2 Free preview
    4. (Included in full purchase)

      CodeChallenge: Riemann approximations 2/2

    5. (Included in full purchase)

      Net vs. total area

    6. (Included in full purchase)

      CodeChallenge: Net and total area

    7. (Included in full purchase)

      Lebesgue integration

    8. (Included in full purchase)

      CodeChallenge: Lebesgue in Blefuscu 1/2

    9. CodeChallenge: Lebesgue in Blefuscu 2/2 Free preview
    10. (Included in full purchase)

      Calculart! Riemannesque animations

  5. 5

    Tangent: Edutainment

    1. (Included in full purchase)

      Pros and cons of edutainment

  6. 6

    Integrating functions

    1. Section summary and goals Free preview
    2. (Included in full purchase)

      How to integrate (formally and practically)

    3. (Included in full purchase)

      Integrating polynomials

    4. (Included in full purchase)

      More on sympy integration

    5. (Included in full purchase)

      Constant of integration: What's with the C?

    6. (Included in full purchase)

      Integration constants in Python

    7. CodeChallenge: Code the initial value problem Free preview
    8. (Included in full purchase)

      CodeChallenge: infinite practice problems

    9. (Included in full purchase)

      Linearity of integration

    10. (Included in full purchase)

      CodeChallenge: Explore linearity 1/2

    11. (Included in full purchase)

      CodeChallenge: Explore linearity 2/2

    12. (Included in full purchase)

      Numerical integration in scipy

    13. (Included in full purchase)

      Integrating sine and cosine

    14. (Included in full purchase)

      Transcendental functions (exp and log)

    15. (Included in full purchase)

      What about product, quotient, and chain rules?

    16. (Included in full purchase)

      Calculart! Whispy sine lines

    17. (Included in full purchase)

      Calculating net and total area

    18. (Included in full purchase)

      CodeChallenge: Net and total areas algorithm

    19. (Included in full purchase)

      Integrating even and odd functions

    20. (Included in full purchase)

      Create practice problems with ChatGPT

    21. (Included in full purchase)

      Reversing the limits of integration

    22. (Included in full purchase)

      Calculart! Create "music" from equations

  7. 7

    Tangent: How can we know anything in math?

    1. (Included in full purchase)

      Mathematical epistomology

  8. 8

    Improper integrals

    1. Section summary and goals Free preview
    2. (Included in full purchase)

      What are improper integrals?

    3. (Included in full purchase)

      Infinite intervals

    4. CodeChallenge: Convergence and divergence Free preview
    5. (Included in full purchase)

      Two infinite bounds

    6. (Included in full purchase)

      CodeChallenge: Improper trig integrals

    7. (Included in full purchase)

      Functions with discontinuities

    8. (Included in full purchase)

      CodeChallenge: approaching real infinity

  9. 9

    Integration techniques

    1. Section summary and goals Free preview
    2. (Included in full purchase)

      U-substitution

    3. (Included in full purchase)

      Integration by parts

    4. (Included in full purchase)

      Partial fractions

    5. CodeChallenge: Partial fractions algorithm Free preview
    6. (Included in full purchase)

      ChatGPT as your integration tutor

    7. (Included in full purchase)

      CodeChallenge: Fun with functions

  10. 10

    Tangent: life as a function to integrate

    1. (Included in full purchase)

      The good life, according to calculus

  11. 11

    Applications in geometry

    1. Section summary and goals Free preview
    2. (Included in full purchase)

      Area between two curves

    3. (Included in full purchase)

      CodeChallenge: Numerical approximations

    4. (Included in full purchase)

      Calculart: A vase, two faces, or just math?

    5. (Included in full purchase)

      Parametric curves

    6. Curve (arc) length: theory Free preview
    7. (Included in full purchase)

      Curve (arc) length: practice (Python)

    8. (Included in full purchase)

      CodeChallenge: approximating curve lengths

    9. (Included in full purchase)

      Calculart: Lissajous's hypotrochoids

    10. (Included in full purchase)

      Creating solids of revolution

    11. (Included in full purchase)

      Measuring volumes of solids

    12. (Included in full purchase)

      Measuring surfaces of solids

    13. (Included in full purchase)

      CodeChallenge: Approximating volumes and surfaces

  12. 12

    Applications in statistics

    1. Section summary and goals Free preview
    2. (Included in full purchase)

      Probability density functions (pdfs)

    3. (Included in full purchase)

      CodeChallenge: Creating pdfs

    4. (Included in full purchase)

      Cumulative distribution functions (cdfs)

    5. (Included in full purchase)

      CodeChallenge: cdfs and pdfs

    6. (Included in full purchase)

      Calculating probabilities

    7. CodeChallenge: Calculating probabilities in python Free preview
  13. 13

    Tangent: is studying math pointless?

    1. (Included in full purchase)

      Why learn math if you'll forget it?

  14. 14

    Multivariable integration

    1. Section summary and goals Free preview
    2. (Included in full purchase)

      Multivariable indefinite integrals

    3. (Included in full purchase)

      CodeChallenge: partial and double integrals

    4. (Included in full purchase)

      Multivariable definite integrals

    5. (Included in full purchase)

      CodeChallenge: numerical double integrals

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