AP Calculus AB · Summary

Unit 6 summary

1. A Riemann sum adds width times height over each piece; in a table, use the real widths between listed inputs.
2. Left and right sums are too low or too high depending on whether f increases or decreases; trapezoidal and midpoint sums depend on concavity.
3. A definite integral is the limit of Riemann sums: area above the x-axis minus area below it, in units of f times units of x.
4. $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$; with an upper limit $u(x)$, multiply by $u'(x)$.
5. $\int_a^b f(x)\,dx = F(b) - F(a)$ for any antiderivative F, and the amount at time b is the starting amount plus the integral of the rate.
6. Every indefinite integral needs + C; use an initial condition to find C.
7. In substitution, pick u as the inner function, rewrite du, and change the limits for a definite integral.