Unit 3 of 7
Structure, limits and approximation
- 4 chapters
- Chapters 11 to 14
Some integrals need more than an antiderivative. Symmetry, a limit over an infinite interval, Simpson's rule with an error bound and a power series each give a value and the reason it holds.
Over a symmetric interval an odd term cancels and an even term doubles, so the parity of each term is read before any calculation. An integral over an infinite interval is a limit of ordinary integrals, and the integral of x to the power minus p from 1 to infinity converges exactly when p is greater than 1. The integral of e to the minus x squared has no elementary antiderivative, so composite Simpson's rule estimates it and an error bound fixes six decimal places. The last chapter integrates a power series term by term, justified by uniform convergence, and bounds the tail of the sum.
Every chapter will be part of one long video. The video is coming soon, and each chapter page already has the definitions and an example for every key term.
Symmetry, improper integrals, Simpson's rule and power series
Read the symmetry before calculating
How odd and even functions behave on a symmetric interval, why an odd integrand integrates to zero and an even one to twice its right half, and how parity splits a mixed integrand before any antiderivative.
Chapter 11
When an infinite region has finite area
How an integral over an infinite interval is defined as a limit, when it is convergent or divergent, and the p-test, by which the integral of x to the power minus p from 1 to infinity converges exactly when p exceeds 1.
Chapter 12
Approximate an integral with a controlled error
How composite Simpson's rule estimates a definite integral with quadratic interpolants, and how Simpson's error bound with a fourth derivative bound fixes the integral of e to the minus x squared to six decimals.
Chapter 13
Integrate an infinite expansion
How to evaluate an integral by term-by-term integration of a power series, using the exponential series, uniform convergence, a removable singularity at an endpoint and a geometric tail bound on the error.
Chapter 14