What's Next?
Congratulations
You have completed all 15 lessons of Calculus 2. You can now implement arc length, surface area, improper integrals, partial sums, Taylor polynomials, parametric arc length, polar area, and curvature — all from scratch in C.
What to Explore Next
- Multivariable Calculus -- Partial derivatives, gradient descent, Lagrange multipliers, double and triple integrals, line integrals, Stokes' theorem
- Differential Equations -- Apply integration and series directly to ODEs: separation of variables, series solutions, Laplace transforms
- Numerical Analysis -- Adaptive step-size quadrature, Gaussian quadrature, Chebyshev approximation — the professional tools built on these ideas
- Complex Analysis -- Taylor and Laurent series in the complex plane, residue theorem, conformal mappings
Key Formulas
| Concept | Formula |
|---|---|
| Arc length | ∫_a^b √(1 + f'²) dx |
| Surface area | 2π ∫_a^b f(x)√(1+f'²) dx |
| Geometric series | a/(1-r) for ` |
| Taylor polynomial | Σ f^(k)(a)/k! · (x-a)^k |
| Polar area | ½ ∫_a^b r(θ)² dθ |
| Curvature | ` |
References
- Calculus by James Stewart — the standard university textbook (Chapters 8–12)
- Numerical Recipes in C — the practitioner's guide to numerical algorithms
- 3Blue1Brown: Essence of Calculus — visual intuition for the concepts