Unit 6: Integration and Accumulation of Change
Work through these 14 topics in order. Clear the gatekeeper and the rest of the course opens up.
6.1 Accumulation of change
If you know the RATE at which something changes, adding up (accumulating) that rate over time gives the TOTAL change.
6.2 Approximating area with Riemann sums
To estimate the area under a curve, slice the region into rectangles and add their areas.
6.3 Riemann sums and definite-integral notation
As the rectangles get infinitely thin, the Riemann sum becomes exact — that limit is the definite integral.
6.4 The FTC and accumulation functions
An accumulation function g(x)=\int_a^x f(t)\,dt builds up area as x moves right.
6.5 Behaviour of accumulation functions
Because g'=f, the shape of an accumulation function g is read from f: g rises where f>0, has extrema where f changes sign, and is concave up where f is increasing.
6.6 Properties of definite integrals
Definite integrals behave like sums: they split over adjoining intervals, pull out constants, add over sums, and flip sign when you reverse the limits.
6.7 Evaluating definite integrals with the FTC
To get the exact area, find an antiderivative and subtract its values at the two limits — no rectangles needed.
6.8 Antiderivatives and indefinite integrals
Antidifferentiation reverses the derivative.
6.9 Integration by u-substitution
Substitution reverses the chain rule: spot an inner function and its derivative, rename the inner function u, and the integral simplifies.
6.10 Long division & completing the square (BC)
When a rational integrand is “top-heavy” or leads to an inverse-trig form, reshape it first: long-divide improper fractions, or complete the square in a denominator.
6.11 Integration by parts (BC)
For a product of two different kinds of function, trade the integral for an easier one using the reverse of the product rule.
6.12 Integration with linear partial fractions (BC)
A rational function with a factorable denominator can be split into simpler fractions, each easy to integrate as a logarithm.
6.13 Improper integrals (BC)
When a limit of integration is infinite (or the integrand blows up), define the integral as a LIMIT and see whether it converges to a finite value.
6.14 Selecting an antidifferentiation technique
Read the form of the integrand first, then pick the tool: basic rule, u-substitution, or (BC) parts / partial fractions / a reshaping step.