Unit 6 Summary (College Board AP® Calculus AB): Study Guide

Integration & accumulation of change summary

Key definitions

• $F\left(x\right)$ is an antiderivative of the function $f\left(x\right)$ if $F\text{'}\left(x\right)=f\left(x\right)$

• An indefinite integral of a function $f$ is denoted $\int f\left(x\right)dx$

• The accumulation of change of a function $f\left(x\right)$ starting at $x=a$ is $F\left(x\right)={\int }_a^{x}f\left(t\right)dt$

• The left Riemann sum to approximate ${\int }_a^{b}f\left(x\right)dx$ is $\sum _{i=1}^n\left(x_i-x_{i-1}\right)·f\left(x_{i-1}\right)$

• The right Riemann sum to approximate ${\int }_a^{b}f\left(x\right)dx$ is $\sum _{i=1}^n\left(x_i-x_{i-1}\right)·f\left(x_i\right)$

• The midpoint Riemann sum to approximate ${\int }_a^{b}f\left(x\right)dx$ is $\sum _{i=1}^n\left(x_i-x_{i-1}\right)·f\left(\frac{x_{i-1}+x_i}{2}\right)$

• The trapezoidal sum to approximate ${\int }_a^{b}f\left(x\right)dx$ is $\sum _{i=1}^n\left(x_i-x_{i-1}\right)·\frac{f\left(x_{i-1}\right)+f\left(x_i\right)}{2}$

• The definite integral can be defined as a limit of Riemann sums

  • ${\int }_a^{b}f\left(x\right)dx=\underset{\max \mathrm{\Delta }{x}_i\to 0}{\lim }\sum _{i=1}^{n}f\mathit{\left(}{x}_i^{\mathit{\ast }}\mathit{\right)}\mathrm{\Delta }{x}_i$

Key theorems

• The first fundamental theorem of calculus

  • If $f$ is continuous on $\left[a,b\right]$ and $F$ is an antiderivative of $f$

  • then ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$

• The second fundamental theorem of calculus

  • If $f$ is continuous on an interval containing $a$, then for values of $x$ in that interval

    • $F\left(x\right)={\int }_a^{x}f\left(t\right)dt$ is an antiderivative of $f$

    • $\frac{d}{dx}\left({\int }_a^{x}f\left(t\right)dt\right)=f\left(x\right)$

Key formulas

• The integral of a function multiplied by a constant: If $k$ is a constant

  • $\int k·f\left(x\right)dx=k·\int f\left(x\right)dx$

• The integral of the sum or difference of two functions:

  • $\int \left(f\left(x\right)\pm g\left(x\right)\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

• The definite integral at a point:

  • ${\int }_a^{a}f\left(x\right)dx=0$

• The change of limits of a definite integral:

  • ${\int }_b^{a}f\left(x\right)dx=-{\int }_a^{b}f\left(x\right)dx$

• Splitting a definite interval at a point: If $a<c<b$

  • ${\int }_a^{b}f\left(x\right)dx={\int }_a^{c}f\left(x\right)dx+{\int }_c^{b}f\left(x\right)dx$

• Indefinite integrals of functions

• If $f$ is continuous on an interval containing $a$, then for values of $x$ in that interval

  • $f\left(x\right)=f\left(a\right)+{\int }_a^{x}f\left(t\right)dt$

• If $f$ is continuous on an interval containing $a$ and $g$ is differentiable, then for values of $x$ in that interval

  • $\frac{d}{dx}\left({\int }_a^{g\left(x\right)}f\left(t\right)dt\right)=f\left(g\left(x\right)\right)·g\text{'}\left(x\right)$

• The reverse chain rule is $\int f\text{'}\left(g\left(x\right)\right)·g\text{'}\left(x\right)dx=f\left(g\left(x\right)\right)+C$

• Integration by substitution can be used to write an integral${\int }_{x_1}^{x_2}f\left(x\right)dx={\int }_{u_1}^{u_2}g\left(u\right)du$

• Completing the square can be used to derive the following integrals

  • $\int \frac{1}{\sqrt{q-{\left(x-p\right)}^2}}dx=\arcsin \left(\frac{x-p}{\sqrt{q}}\right)+C$

  • $\int \frac{1}{q+{\left(x-p\right)}^2}dx=\frac{1}{\sqrt{q}}\arctan \left(\frac{x-p}{\sqrt{q}}\right)+C$

Key facts

• The criteria for under/overestimates can be summarized in the table below