2.5.1 Area and Riemann sums

What area under a curve means

The area under the graph of $f$ from $x = a$ to $x = b$ is the region between the curve and the $x$ -axis over that stretch. Its units are height units times width units.

That makes area more than geometry. A car at 60 km per hour for 2 hours covers $60 \times 2 = 120$ km. That is the area of a rectangle of height 60 and width 2 under the graph of its speed.

When the speed changes, the region under the speed graph is no longer a rectangle, but its area is still the distance. No school formula gives the area under a curve. So this lesson cuts the region into thin strips, covers each with a rectangle, and adds up the rectangles.

Rectangles under $x^2$

Take $f(x) = x^2$ from 0 to 1. Cut $[0, 1]$ into $n = 4$ strips of equal width $\Delta x = \frac{1 - 0}{4} = \frac{1}{4}$ . Each strip gets a rectangle whose height is the value of $f$ at one point of the strip.

heights taken at	points	heights	total area
left ends	$0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}$	$0, \frac{1}{16}, \frac{1}{4}, \frac{9}{16}$	0.21875
right ends	$\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1$	$\frac{1}{16}, \frac{1}{4}, \frac{9}{16}, 1$	0.46875
midpoints	$\frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}$	$\frac{1}{64}, \frac{9}{64}, \frac{25}{64}, \frac{49}{64}$	0.328125

Each total is the sum of the four heights times the width $\frac{1}{4}$ . For the left ends, $\left(0 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16}\right) \times \frac{1}{4} = \frac{7}{32} = 0.21875$ . The three totals differ, so each is only an estimate of the area.

Writing the sum with Σ

Number the strips $i = 1, 2, \ldots, n$ . Strip $i$ runs from $x_{i-1}$ to $x_i$ , where $x_i = a + i\,\Delta x$ , so $x_0 = a$ and $x_n = b$ . Write $x_i^*$ (read "x i star") for the point where strip $i$ takes its height.

Rule: Riemann sum

\begin{gathered} \Delta x = \frac{b - a}{n}, \quad x_i = a + i\,\Delta x \\[4pt] \sum_{i=1}^{n} f(x_i^*)\,\Delta x \end{gathered}

Each term is one rectangle: height $f(x_i^*)$ times width $\Delta x$ . The left sum takes $x_i^* = x_{i-1}$ , the right sum $x_i^* = x_i$ , and the midpoint sum the middle of the strip, $\frac{x_{i-1} + x_i}{2}$ .

Every term has the same width, so $\Delta x$ comes out in front: $\sum f(x_i^*)\,\Delta x = \Delta x \sum f(x_i^*)$ . That is the constant multiple rule of lesson 0.2.2, and it is why each total above is "add the heights, then multiply by $\frac{1}{4}$ ".

0, 1, \ldots, n - 1

import numpy as np def riemann(f, a, b, n, rule): dx = (b - a) / n i = np.arange(n) if rule == "left": x = a + i * dx elif rule == "right": x = a + (i + 1) * dx else: x = a + (i + 0.5) * dx return np.sum(f(x)) * dx for rule in ["left", "right", "mid"]: print(rule, riemann(np.square, 0, 1, 4, rule))

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Too large or too small?

For a rising curve, the left sum is too small and the right sum too large. For a falling curve it is the other way round.

On a strip where $f$ rises, the left end is the lowest point. So the left rectangle sits under the curve and misses a sliver of area, while the right rectangle pokes above it. $x^2$ rises on $[0, 1]$ , so the left sum 0.21875 is too small and the right sum 0.46875 too large.

$\frac{1}{x}$ falls on $[1, 2]$ . With 4 strips its left sum is 0.759524 and its right sum 0.634524. The exact area is $\ln 2 \approx 0.693147$ (lesson 2.5.3 shows why), so now the left sum is too large and the right sum too small.

Rule: Which way the error goes

\begin{gathered} f \text{ rising: left sum} < \text{area} < \text{right sum} \\ f \text{ falling: right sum} < \text{area} < \text{left sum} \end{gathered}

The exact area is trapped between the two sums. A curve that rises and then falls follows neither line. For $\sin x$ on $[0, \pi]$ with 4 strips, both sums are 1.896119, under the exact area 2.

The midpoint sum lands on a side that depends on how the curve bends (lesson 2.4.2). It is too small for $x^2$ (0.328125) and $\frac{1}{x}$ (0.69122), which bend up. It is too large for $\sin x$ on $[0, \pi]$ (2.052344), which bends down.

The gap between the left and right sums has a simple formula. Both use the heights at $x_1, \ldots, x_{n-1}$ ; only the two ends differ. So the right sum minus the left sum is $\Delta x\,\big(f(b) - f(a)\big)$ : for $x^2$ with 4 strips, $\frac{1}{4} \times (1 - 0) = 0.25$ .

Try it in the explorer. It starts with the four left rectangles of the table: switch to right ends and midpoints and compare the totals. Then pick $\frac{1}{x}$ on $[1, 2]$ and watch the left and right sums swap sides, and finally raise $n$ towards 100.

Explore

Add rectangles and change where each takes its height. The total area of the rectangles closes in on the exact area under the curve.

Rectangles n = 4

Width of each: 1 ÷ 4 = 0.25
Total of 4 rectangles: 0.21875
Exact area: 1/3 = 0.333333
Error: −0.114583 (an underestimate)

Thinner strips, smaller errors

As the strips get thinner, all three sums close in on the same number. For $x^2$ on $[0, 1]$ :

strips $n$	4	10	100	10000
left sum	0.21875	0.285	0.32835	0.333283
right sum	0.46875	0.385	0.33835	0.333383
midpoint sum	0.328125	0.3325	0.333325	0.333333

The left and right errors shrink in step with $\Delta x$ . Ten times as many strips cut the left error about ten times, from 0.04833 at $n = 10$ to 0.00498 at $n = 100$ . The midpoint error shrinks with $\Delta x^2$ : ten times as many strips cut it a hundred times, from 0.00083 to 0.0000083.

So the midpoint sum is the one to use when you estimate an area in code. Lesson 2.8.2 builds Simpson's rule, which is faster still.

The same limit by algebra

For the right sum of $x^2$ on $[0, 1]$ , the points are $x_i = \frac{i}{n}$ and the width is $\frac{1}{n}$ . Pulling $\frac{1}{n^3}$ out in front leaves the sum of the first $n$ squares.

\begin{aligned} R_n &= \sum_{i=1}^{n} \left(\frac{i}{n}\right)^2 \frac{1}{n} \\ &= \frac{1}{n^3}\sum_{i=1}^{n} i^2 \\ &= \frac{(n + 1)(2n + 1)}{6n^2} \end{aligned}

The last line uses the formula $\sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6}$ , which the Proof Gym proves by induction. For $n = 4$ it gives $\frac{4 \times 5 \times 9}{6} = 30$ , the sum $1 + 4 + 9 + 16$ of lesson 0.2.2. Then $R_4 = \frac{45}{96} = 0.46875$ , the right sum in the table.

As $n \to \infty$ , the top $(n + 1)(2n + 1)$ grows like $2n^2$ and the bottom is $6n^2$ , so $R_n \to \frac{2}{6} = \frac{1}{3}$ (the biggest power wins, lesson 2.1.3). The error is $R_n - \frac{1}{3} = \frac{3n + 1}{6n^2}$ , about $\frac{1}{2n}$ : it halves each time $n$ doubles.

n

import numpy as np def riemann(f, a, b, n, rule): dx = (b - a) / n i = np.arange(n) offset = {"left": 0, "right": 1, "mid": 0.5}[rule] return np.sum(f(a + (i + offset) * dx)) * dx for n in [4, 10, 100, 10000]: sums = [round(riemann(np.square, 0, 1, n, r), 6) for r in ["left", "mid", "right"]] print(n, *sums, round((n + 1) * (2 * n + 1) / (6 * n**2), 6))

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The definite integral

When the sums for every choice of point settle on one number, that number is the definite integral. They do for every continuous $f$ (lesson 2.1.2) on a closed interval $[a, b]$ .

Rule: Definite integral

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x

The sign $\int$ is a stretched S, for "sum". $f(x)$ is the integrand, $a$ and $b$ are the lower and upper limits of integration, and $dx$ recalls the width $\Delta x$ . So $\int_0^1 x^2\,dx = \frac{1}{3}$ .

A definite integral is a number, in height units times width units. For the car, $\int_0^2 60\,dt = 120$ km.

The letter inside is a placeholder, like the index of a sum in lesson 0.2.2: $\int_0^1 x^2\,dx$ and $\int_0^1 t^2\,dt$ are the same number. If $f$ blows up inside $[a, b]$ , as $\frac{1}{x}$ does at 0, the sums need not settle. Lesson 2.6.3 deals with such integrals.

Signed area

Where $f$ is negative, the integral counts the area as negative. Below the axis, each height $f(x_i^*)$ is negative, so each term $f(x_i^*)\,\Delta x$ is negative too.

So the integral adds the area above the axis and subtracts the area below. This is the net, or signed, area. The total area counts every piece as positive.

The region between $y = x$ and the axis from $-1$ to 2. The integral is the net area, $2 - 0.5 = 1.5$ ; the total area is $2 + 0.5 = 2.5$ .

Both triangles can be checked with $\frac{1}{2} \times \text{base} \times \text{height}$ : $\frac{1}{2} \times 2 \times 2 = 2$ above the axis, and $\frac{1}{2} \times 1 \times 1 = 0.5$ below it. So $\int_{-1}^{2} x\,dx = 1.5$ .

A net area of zero does not mean there is no area. On $[0, 2\pi]$ , the dip of $\sin x$ below the axis is a mirror image of the hump above it, so they cancel: $\int_0^{2\pi} \sin x\,dx = 0$ . Each piece has area 2 (the explorer gives this exact area for $\sin x$ on $[0, \pi]$ ), so the total area is 4.

Rule: Net and total area

\begin{gathered} \text{net} = \text{area above} - \text{area below} \\ \text{total} = \text{area above} + \text{area below} \end{gathered}

The integral $\int_a^b f(x)\,dx$ is the net area. The total area is the integral of $|f(x)|$ , which flips the parts below the axis up. Lesson 2.5.3 finds both exactly.

\sin x

import numpy as np def mid(f, a, b, n=1200): dx = (b - a) / n return np.sum(f(a + (np.arange(n) + 0.5) * dx)) * dx def line(x): return x print(round(mid(line, -1, 2), 6), round(mid(lambda x: np.abs(line(x)), -1, 2), 6)) print(round(mid(np.sin, 0, 2 * np.pi), 6), round(mid(lambda x: np.abs(np.sin(x)), 0, 2 * np.pi), 4))

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Common mistakes

Height times width. Each term is $f(x_i^*)\,\Delta x$ . Adding the heights alone forgets the width.
The width is $\frac{b - a}{n}$ , not $\frac{1}{n}$ . With 4 strips on $[0, 2]$ , $\Delta x = 0.5$ .
Count the points. $n$ strips have $n + 1$ edges. The left sum uses the first $n$ of them, the right sum the last $n$ .
"The left sum is too small" holds only for a rising curve. For a falling curve it is too large.
A net area of 0 does not mean no area. $\sin x$ on $[0, 2\pi]$ has net area 0 and total area 4.
A Riemann sum is an estimate. The integral is its limit as $n \to \infty$ .

Riemann sums and integrals

0 of 12 correct

1.
With 6 strips on $[1, 4]$ , what is $\Delta x$ ?
2.
Left sum for $x^2$ on $[0, 1]$ with 4 strips: $(0 + \tfrac{1}{16} + \tfrac{1}{4} + \tfrac{9}{16}) \times \tfrac{1}{4}$
3.
Right sum for $x^2$ on $[0, 3]$ with 3 strips
4.
Midpoint sum for $2x$ on $[0, 2]$ with 2 strips
5.
$\frac{1}{x}$ falls on $[1, 3]$ . Find its right sum with 2 strips (it is too small)
6.
$x^2$ rises on $[0, 2]$ . With 4 strips, by how much does the right sum exceed the left sum?
7.
The left-sum error for $x^2$ on $[0, 1]$ is about $\frac{1}{2n}$ . About how big is it with 50 strips?
8.
A midpoint sum is off by 0.0008 with 10 strips. About how far off is it with 100 strips?
9.
A tap runs at 12 litres per minute for 5 minutes. How many litres is the area under its rate graph?
10.
Find $\int_{-2}^{1} x\,dx$ , the signed area
11.
Find the total area between $y = x$ and the $x$ -axis from $-2$ to 1
12.
Find $\int_0^{2\pi} \sin x\,dx$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

f

Area and Riemann sums

What area under a curve means

Rectangles under $x^2$

Writing the sum with Σ

Rule: Riemann sum

Too large or too small?

Rule: Which way the error goes

Thinner strips, smaller errors

The same limit by algebra

The definite integral

Rule: Definite integral

Signed area

Rule: Net and total area

Common mistakes

Riemann sums and integrals

Summary card

Free further reading

What area under a curve means

Rectangles under x2x^2x2

Writing the sum with Σ

Rule: Riemann sum

Too large or too small?

Rule: Which way the error goes

Thinner strips, smaller errors

The same limit by algebra

The definite integral

Rule: Definite integral

Signed area

Rule: Net and total area

Common mistakes

Summary card

Free further reading

Rectangles under $x^2$