Riemann sums: chasing the area under a curve with thin strips

How to use this simulation

1. Pick a function from “Function f(x)”: x², x³, sin x, eˣ or √x. The black curve is y = f(x). The two orange ticks on the axis mark the ends a and b of the interval you are measuring.
2. The shaded strips are the Riemann sum. “Number of strips, n” sets how many there are, from 1 to 100. Start with n = 4 and see the rectangles stick out above or fall short of the curve.
3. Choose “Which height to use”. Left end and right end take the height of the curve at the edge of each strip. Midpoint takes it in the middle. Trapezoid joins the two edge heights with a slanted top.
4. Move the n slider up. The error between your “Riemann sum” and the “Exact integral” shrinks. Watch which rule shrinks it fastest. Press play to let n sweep upward by itself.
5. Drag a and b to change the interval. Colour tells the sign: blue strips lie above the axis and count as positive, red strips lie below it and count as negative, so the sum is a net area.
6. Read the error. “Error (sum − exact)” keeps its sign: plus means you overshot, minus means you undershot. “Relative error” says how big that is as a percentage of the true value.

Area of a shape with a curvy edge

Rectangles and triangles have formulas. A shape with a curved edge does not. Think of a piece of land bounded by a straight road on one side and a winding river on the other. There is no neat formula for its area. But there is a neat trick: cut the land into thin vertical strips.

Each thin strip is almost a rectangle, and you can find the area of a rectangle. Add the areas up and you have an estimate of the whole. Cut the strips thinner and the estimate improves. That is the entire idea behind integration.

Building a Riemann sum

Pick the interval [a, b] and the number of strips n. Every strip has the same width, Δx = (b − a) / n. For each strip choose a sample point, read the curve’s height there, and make a rectangle that wide and that tall. Its area is height × Δx.

Here is f(x) = x² on [0, 3] with n = 4. The strip width is Δx = 0.75. The exact area is 9, as we will check below. The Riemann sum is the total of the four rectangle areas.

Four ways to pick the height

The rule says where in the strip to read the height. Different choices give different answers, and each has a personality.

• Left end: height at the strip’s left edge. On a rising curve every rectangle sits under the curve, so the sum underestimates.
• Right end: height at the right edge. On a rising curve every rectangle pokes above the curve, so the sum overestimates.
• Midpoint: height in the middle. The overshoot on one side nearly cancels the shortfall on the other, so it is usually much more accurate.
• Trapezoid: average of the left and right sums. The slanted top hugs the curve better than a flat one.

The four rules on one example

All four rules on x² over [0, 3] with only four strips. The left sum is 5.9063 and the right sum is 12.6563, one under and one over the true value 9. Midpoint and trapezoid land much closer.

More strips, smaller error

Now keep the left rule but grow n. The table shows the sum and the error, plus the error multiplied by n. That last column settles near −13.5: so the error shrinks like 1 ÷ n. Double the strips, halve the error.

Above and below the axis: signed area

A rectangle below the x-axis has a negative height, so it contributes a negative area. The Riemann sum, and the definite integral it leads to, are net areas: area above the axis minus area below. That is why the strips are coloured blue above and red below in the simulation.

If you want the plain total area, split the interval where the curve crosses the axis and add the sizes of the pieces.

From the sum to the integral

Let n go to infinity and the strips become infinitely thin. The sum settles on a single number, the definite integral. We rarely compute that limit by hand. Instead we use the antiderivative: if F′(x) = f(x), then the integral from a to b is F(b) − F(a).

The simulation uses exactly this to print “Exact integral”. For x² the antiderivative is x³/3, for x³ it is x⁴/4, for sin x it is −cos x, for eˣ it is eˣ, and for √x it is (2/3)x^(3/2).

Try it in the simulation

Six quick experiments, a minute each.

• Choose x² on [0, 3] with the left rule. Slide n from 1 to 100. What does the error do each time you double n?
• Switch to the midpoint rule at n = 4. How does its error compare to the left rule at n = 4? And at n = 8?
• Choose sin x and the right rule. Is the sum above or below the exact value? Why does it change if you move the interval to include the downward part of the wave?
• Choose x³ and set a = −2 and b = 2. The strips turn red on the left. What is the exact integral? Why?
• Choose √x with the left rule. The curve is very steep at 0, so convergence is slow. How many strips does it take to get the error under 0.05?
• Choose eˣ and compare left and right at n = 10. The exact answer sits between them. How close is the trapezoid sum?

Solved problems

Eight common questions. Every number below is computed with the same functions as the simulation.

Common mistakes

Four slips that cost marks every year.

• Using n points instead of n strips. Δx is (b − a) divided by the number of strips.
• Forgetting to multiply by Δx. Adding up the heights alone is not an area.
• Reading the right ends in the left rule (or the other way round). Write the sample points down first.
• Treating a negative integral as an error. It simply means more area below the axis than above.

Where Riemann sums are used

Distance travelled is the area under a velocity-time graph, and a car’s trip computer adds thin time-strips to get it. Total rainfall, the energy used by a house, the work done by a changing force, the amount of medicine in the blood: all areas under curves. Computers use these rules, in smarter forms, whenever a function has no neat antiderivative.

Exam corner

Typical SSC higher-math, HSC and first-year questions on this topic.

• Estimate an area using given rectangles or trapezoids and state whether it over- or under-estimates.
• Write a sum in Σ notation for given n, a and b.
• Evaluate a definite integral with the antiderivative and compare it to a sum.
• Say how the error changes when n doubles.

Revision summary

Δx = (b − a) / n. Sum of height × Δx. Left, right, midpoint, trapezoid. Error shrinks as n grows: like 1/n for left and right, like 1/n² for midpoint and trapezoid. Limit of the sums is the definite integral, F(b) − F(a).

Frequently asked questions