AP Calculus AB · Summary

Unit 8 summary

1. Average value is $\frac{1}{b - a}\int_a^b f(x)\,dx$. It is not the mean of the endpoint values, and not the average rate of change.
2. Displacement is $\int v\,dt$; total distance is $\int |v|\,dt$. Position at b is the starting position plus the displacement.
3. Amount at b = amount at a + the integral of the net rate. The amount is greatest or least where the net rate changes sign or at an endpoint.
4. Area between curves is the integral of top minus bottom between the intersections. Split where the curves cross, or slice in y with right minus left.
5. Volume is the integral of the slice area: $s^2$ for squares, $\frac{\pi s^2}{8}$ for semicircles, $\pi R^2$ for discs and $\pi(R^2 - r^2)$ for washers.
6. Every radius is a distance from the axis of revolution. About $y = k$, use $|f(x) - k|$, and square each radius before subtracting.