Riemann sum explorer

Pick a function and a rule, then move the slider to change the number of strips $n$ . The explorer adds up the strips and sets the total next to the exact value of the integral. Strips above the $x$ axis count as positive and strips below it count as negative, so the sum estimates signed area.

Function and interval

Rule

LeftRightMidpointTrapezoid

Subintervals n4

Δx = 0.25

Above the axis, counted as positive
Below the axis, counted as negative

Left sum: 0.218750
Exact integral: 0.333333Equals 1/3
Error: −0.114583The sum is too small
Simpson's rule: 0.333333Error 0.000000

Things to try

Start with $x^2$ on $[0, 1]$ and $n = 4$ . The left sum is $0.21875$ and the right sum is $0.46875$ , so the exact area $\frac{1}{3}$ lies between them. That holds for every $n$ , because $x^2$ is increasing there.
Stay with $x^2$ and go from $n = 4$ to $n = 8$ . The left sum error goes from about $-0.1146$ to about $-0.0599$ , roughly half. The midpoint error goes from $-0.005208$ to $-0.001302$ , exactly a quarter.
Compare the midpoint and trapezoid errors for $x^2$ , $e^x$ or $\frac{1}{x}$ . The trapezoid error is close to twice the midpoint error and has the opposite sign. For $x^2$ the ratio is exactly $-2$ at every $n$ .
With $\sin x$ on $[0, \pi]$ or the half circle $\sqrt{1 - x^2}$ on $[-1, 1]$ , the left, right and trapezoid sums agree for every $n$ . The function is $0$ at both ends, so the first left rectangle and the last right rectangle add nothing.
Pick $x^3 - x$ on $[-1, 1.5]$ with the left rule and $n = 4$ . The sum is $-0.073242$ even though the integral is $\frac{25}{64} = 0.390625$ . The negative strips between $0$ and $1$ outweigh the positive ones for $n = 2$ , $3$ and $4$ , and the left sum turns positive at $n = 5$ .
Watch Simpson’s rule on the three polynomials. For $x^2$ , $4 - x^2$ and $x^3 - x$ it gives the exact value at every even $n$ .
On the half circle Simpson’s rule loses to the midpoint rule at every even $n$ . The slope of $\sqrt{1 - x^2}$ grows without bound near $x = \pm 1$ , and the error estimates behind Simpson’s rule need a smooth function.
For $e^x$ on $[0, 1]$ , find the smallest $n$ that brings the midpoint error under $0.0001$ . It is $n = 27$ .

What you are seeing

The interval $[a, b]$ is cut into $n$ pieces of equal width with the points $x_i$ ,

\begin{gathered} \Delta x = \frac{b - a}{n}, \\[0.8em] x_i = a + i \, \Delta x \quad (i = 0, 1, \ldots, n). \end{gathered}

The left, right and midpoint rules sample $f$ at one point of each piece and add the rectangles,

\begin{aligned} L_n &= \sum_{i=1}^{n} f(x_{i-1}) \, \Delta x, \\[0.8em] R_n &= \sum_{i=1}^{n} f(x_i) \, \Delta x, \\[0.8em] M_n &= \sum_{i=1}^{n} f\!\left(\frac{x_{i-1} + x_i}{2}\right) \Delta x. \end{aligned}

The trapezoid rule joins neighboring points of the graph with straight segments. It equals the average of the left and right sums,

\begin{aligned} T_n &= \frac{\Delta x}{2} \bigl[ f(x_0) + 2f(x_1) + \cdots \\[0.4em] &\qquad + 2f(x_{n-1}) + f(x_n) \bigr] \\[0.8em] &= \frac{L_n + R_n}{2}. \end{aligned}

When $n$ is even, Simpson’s rule fits a parabola through each three neighboring points,

\begin{aligned} S_n = \frac{\Delta x}{3} \bigl[ & f(x_0) + 4f(x_1) + 2f(x_2) + \cdots \\[0.4em] & + 4f(x_{n-1}) + f(x_n) \bigr]. \end{aligned}

A product $f(x) \, \Delta x$ with $f(x) < 0$ is negative, which is why the strips below the axis are shaded differently and subtract from the total. A trapezoid whose top edge crosses the axis is split there, and each part is shaded by its sign.

The exact value comes from an antiderivative, for example

\int_1^2 \frac{1}{x} \, dx = \ln 2 \approx 0.693147.

The error shown is the sum minus the exact value, so a positive error means the sum is too big. Left and right sum errors shrink roughly like $\frac{1}{n}$ . Midpoint and trapezoid errors shrink like $\frac{1}{n^2}$ , and Simpson errors like $\frac{1}{n^4}$ for smooth functions.

These sums are built up in The Area Problem, Partitions and Riemann Sums and Integral Approximation Rules, and the formulas are on the calculus reference sheet. The videos are on the Calculus course page. For a first look at rectangle sums without calculus, see Area Approximation with Rectangles on the Precalculus course page.