Topic 6.2 Notes – Approximating Areas with Riemann Sums

What a Riemann Sum Is

A definite integral
$$\int_a^b f(x)\,dx$$
represents the net area under $f(x)$ from $a$ to $b$.

A Riemann sum approximates that area by:

• Dividing $[a,b]$ into subintervals (a partition)
• Approximating the curve on each subinterval with a simple shape
• Adding the areas

Key pieces you must recognize:

• $n$ = number of subintervals
• $\Delta x$ = width of each subinterval
  • Uniform: $\Delta x = \frac{b-a}{n}$
  • Nonuniform: widths may vary
• $x_i^*$ = sample point in each interval

General form:
$$\sum f(x_i^*)\,\Delta x$$

As $n$ increases, the approximation improves. In the limit, it becomes the exact integral.

You must be able to approximate from:

• A graph
• A table of values
• An equation
• A verbal rate description (very common on FRQs)

The Four Types of Riemann Sums

They all approximate area. The only difference is how the height is chosen.

Left Riemann Sum $L_n$

Height comes from the left endpoint.

$$L_n = \sum_{i=1}^{n} f(x_{i-1})\Delta x$$

If the function is increasing, rectangles sit mostly below the curve.

Right Riemann Sum $R_n$

Height comes from the right endpoint.

$$R_n = \sum_{i=1}^{n} f(x_i)\Delta x$$

If the function is increasing, rectangles sit mostly above the curve.

Midpoint Riemann Sum $M_n$

Height comes from the midpoint of each subinterval.

$$M_n = \sum f(\text{midpoint})\Delta x$$

Often more accurate than left or right.

Trapezoidal Sum $T_n$

Instead of rectangles, use trapezoids connecting endpoints.

Area of one trapezoid:
$$\frac{f(x_{i-1}) + f(x_i)}{2}\Delta x$$

Very important shortcut:
$$T_n = \frac{L_n + R_n}{2}$$

That relationship shows up a lot on quizzes.

Visualizing the Differences

How to Compute One From a Formula

Suppose you’re asked for $R_3$ for $f(x)=x^2$ on $[1,4]$.

1. Compute width
   $$\Delta x = \frac{4-1}{3} = 1$$
2. Right endpoints: 2, 3, 4
3. Evaluate
   $f(2)=4$, $f(3)=9$, $f(4)=16$
4. Multiply by width and add
   $$R_3 = (4+9+16)(1) = 29$$

Same logic works with tables. If the widths aren’t equal, multiply each height by its specific interval width before adding.

On calculator-active parts of the AP exam, you may store values in lists. But setup still matters. If your $x$-values are wrong, everything is wrong.

Overestimates and Underestimates

This is where students mix things up.

Increasing or Decreasing → Left vs Right

You’re thinking about whether rectangles sit below or above the curve.

Concavity → Midpoint vs Trapezoidal

Why? Trapezoids use secant lines, and secant lines lie above a concave up curve.

AP questions often ask you to justify using “since $f$ is increasing…” or “since $f$ is concave up…”. Use the correct vocabulary.

Numerical and Verbal Approximations

Sometimes you’re given:

• A table of rate values at specific times
• Unequal time intervals
• A description like “water flows at a rate of 5 gallons per minute…”

Then:

• Multiply each rate by its time width
• Add to approximate total accumulation

Units matter. Rate × time = amount.

Nonuniform partitions are common on FRQs. Don’t assume equal spacing unless told.