Common power sums: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$, $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$
These properties let you break complicated sums into simple solvable parts, which is necessary when working with $n$ approaching infinity, where adding individual terms is impossible.
Exam tip: If a question asks you to take the limit of a sigma expression, you never need to add every term from $i=1$ to $n$ manually. Always use summation properties to group constants and substitute power sums first.
2. Approximating Net Area with Riemann Sums★★☆☆☆⏱ 4 min
A Riemann sum approximates the net area between a continuous function $f(x)$ and the x-axis over a closed interval $[a, b]$. To construct a Riemann sum with equal-width subintervals: split $[a,b]$ into $n$ subintervals, pick a sample point in each subinterval, then calculate and sum the area of each rectangle.
**Left Riemann sum**: $x_i^*$ is the left endpoint of each subinterval
**Right Riemann sum**: $x_i^*$ is the right endpoint of each subinterval
**Midpoint Riemann sum**: $x_i^*$ is the midpoint of each subinterval
A key property: Riemann sums calculate **net area**, which equals the area of the function above the x-axis minus the area of the function below the x-axis. Rectangles where $f(x_i^*)<0$ contribute a negative value to the total sum.
Exam tip: When given a table of values for $f(x)$ (a common AP question), label all subinterval endpoints first before picking sample points, to avoid mixing up left and right endpoints.
3. Definite Integral Notation as a Limit of Riemann Sums★★★☆☆⏱ 3 min
To get the exact net area under $f(x)$ over $[a,b]$, we take the limit of the Riemann sum as $n \to \infty$, which makes $\Delta x \to 0$. This limit is defined as the definite integral of $f(x)$ from $a$ to $b$:
In this notation: $\int$ is the integral sign (from "sum"), $a$ is the lower limit of integration, $b$ is the upper limit, $f(x)$ is the integrand, and $dx$ comes from $\Delta x$, representing the infinitesimal width of each subinterval. The most common AP skill here is converting a limit of a Riemann sum to a definite integral, which follows 3 steps: (1) Identify $\Delta x = \frac{c}{n}$, so $b-a = c$; (2) Match $x_i^* = a + i\Delta x$ to find the lower bound $a$; (3) Calculate $b = a + c$ and write $\int_a^b f(x) dx$.
Exam tip: Never assume the lower bound of integration is 0 just because the sum starts at $i=1$. Many AP trap questions use non-zero lower bounds to test your ability to match $x_i^*$ to $a + i\Delta x$.
4. AP-Style Worked Practice Problems★★★☆☆⏱ 3 min
Common Pitfalls
Why: Confusing the coefficient of $i/n$ in $x_i^*$ with $\Delta x$; the term multiplied by $1/n$ outside the function is always $\Delta x$
Why: Memorizing over/under estimates incorrectly instead of reasoning from the function's shape
Why: Forgetting that subintervals are $[0,2], [2,4], [4,6]$, so midpoints are at $1, 3, 5$, not sequential integers starting at 1
Why: Confusing terms that depend on $i$ (which change for each term) with constants that do not depend on $i$
Why: Forgetting that Riemann sums calculate net area, not total geometric area