Approximating Area

How much area lies under $y = x^2$ between $x = 0$ and $x = 2$? There's no geometry formula for a curved region, so we approximate it with rectangles.

Riemann sums

Split $[a, b]$ into $n$ strips of equal width

$$\Delta x = \frac{b - a}{n}$$

with endpoints $x_0 = a,\ x_1 = a + \Delta x,\ \dots,\ x_n = b$. On each strip build a rectangle whose height is the function's value at one sample point, then add up the areas:

Rule	Height of the $i$-th rectangle	Sum
left	value at the strip's left end	$L_n = \sum_{i=0}^{n-1} f(x_i)\,\Delta x$
right	value at the strip's right end	$R_n = \sum_{i=1}^{n} f(x_i)\,\Delta x$
midpoint	value at the strip's middle	$M_n = \sum_{i=1}^{n} f\!\left(\frac{x_{i-1} + x_i}{2}\right)\Delta x$

Example: $f(x) = x^2$ on $[0, 2]$ with $n = 4$

$\Delta x = 0.5$, and the endpoints are $0, 0.5, 1, 1.5, 2$.

• Left: $L_4 = 0.5\,(0 + 0.25 + 1 + 2.25) = 1.75$
• Right: $R_4 = 0.5\,(0.25 + 1 + 2.25 + 4) = 3.75$
• Midpoint (at $0.25, 0.75, 1.25, 1.75$): $M_4 = 0.5\,(0.0625 + 0.5625 + 1.5625 + 3.0625) = 2.625$

The true area turns out to be $\frac{8}{3} \approx 2.667$ (you'll be able to show this in section 07.04). The midpoint sum is much closer than either endpoint sum.

Too high or too low?

• If $f$ is increasing, left rectangles sit under the curve (underestimate) and right rectangles poke above it (overestimate).
• If $f$ is decreasing, it's the other way around.
• Averaging the two gives the trapezoid rule, $T_n = \frac{L_n + R_n}{2}$. Above, $T_4 = 2.75$.

Sigma notation

$\sum_{i=1}^{n} a_i$ means $a_1 + a_2 + \dots + a_n$. Constants pull out and sums split, and two formulas come up again and again:

$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}, \qquad \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$

Rates from a table

The same idea works on data. If a car's velocity is measured every $2$ seconds, then velocity × time on each interval estimates the distance covered then, and adding the pieces estimates the total distance.