Integral as area: Riemann sums

Riemann sums

How do thin rectangles add up to the exact area under a curve?

Fill the region under y = x²/4 + 1/2 with rectangles, then add more of them. The total closes in on the integral, and the error shrinks like 1/n.

Drag the ringed points on the board.

High school
University
Riemann sum
Definite integral
Limit of a sum
Fundamental theorem

n

b

3.00

L_n \to \int_0^b f(x)\,dx

$n$: 4
$\Delta x$: 0.750
$L_n$: 2.9766
$\int_0^b f\,dx$: 3.7500
$\text{error}$: 0.7734

Error under 0.01not yet

Guided tour

1Two rough blocks
With $n=2$ on $[0,3]$ each rectangle takes its height from its left edge. Because the curve rises, every block falls short: the sum is $2.34$ against an exact area of $3.75$ .
2Four blocks
At $n=4$ the sum is $2.98$ . The gaps between the staircase and the curve are the error, and each gap is a thin sliver.
3Sixteen blocks
At $n=16$ the sum is $3.54$ and the error is about $0.21$ . Quadrupling $n$ cut the error by roughly four.
4The limit is the integral
At $n=60$ the staircase is almost the curve and the sum is $3.69$ . As $n\to\infty$ it reaches $\int_0^3 f(x)\,dx = 3.75$ exactly.

Explanation

1
Slice, approximate, add
Area under a straight line is easy; area under a curve is not. So cut $[a,b]$ into $n$ strips of width $\Delta x=\frac{b-a}{n}$ and pretend each strip is a rectangle. Its height is the function value at one sample point $x_i$ in the strip.
$L_n=\sum_{i=0}^{n-1} f(x_i)\,\Delta x$
This is a Riemann sum. On the board the sample is the left edge of each strip, so $L_n$ is a left sum.
2
The definite integral
More strips make thinner slivers of error. The definite integral is the value the sums approach:
$\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=0}^{n-1} f(x_i)\,\Delta x$
For a continuous function the limit is the same whichever sample point you use: left, right or midpoint.
3
Computing the exact area
The fundamental theorem of calculus replaces the limit with an antiderivative. For $f(x)=\frac{x^2}{4}+\frac12$ , an antiderivative is $F(x)=\frac{x^3}{12}+\frac{x}{2}$ , so
$\int_0^b f(x)\,dx = F(b)-F(0)=\frac{b^3}{12}+\frac{b}{2}$
At $b=3$ that is $\frac{27}{12}+\frac32=3.75$ , the number the readout shows.
4
Why a left sum undershoots here
On an increasing function the left edge is the lowest point of each strip, so every rectangle sits under the curve and $L_n$ is too small. A right sum would overshoot by about the same amount. Their average, the trapezoid rule, is far more accurate.

Check yourself

Q01

On $[0,3]$ with $n=6$ strips, what is $\Delta x$ ?

Q02

For an increasing function, how does the left Riemann sum compare with the integral?

Q03

What is $\int_0^2 \left(\frac{x^2}{4}+\frac12\right)dx$ ?

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