The area under a curve

What you will be able to do

After this lesson you can

• find an area under a curve by adding narrow pieces,
• compare rectangles and trapezoids and say how their errors behave, and
• find a distance from a speed graph with np.trapezoid.

Why areas?

In chapter 2, you found a distance from a speed graph as the area under the graph. This idea is very general. The area under a force graph is work. The area under a power graph is energy. The area under a curve of the rate of flow is the volume that flowed. In calculus, such an area is a definite integral. We learn to find it by calculation.

Cutting into pieces

The area under $f(x) = x^2$ from 0 to 1 is $\tfrac{1}{3}$ (you can find it by integration). Cut the interval into $n$ equal pieces. Then replace the curved top of each piece by something easy.

• Left rectangles: the height is the value at the left end of each piece. For a rising curve, each rectangle is below the curve, and the sum is too small.
• Right rectangles: the height is the value at the right end. For a rising curve, it is too big.
• Trapezoids: join the two ends by a straight line. The area of a trapezoid is the average of the two heights times the width. It is close to the curve.

Predict first. Which sum will be too small, and which too big? Which will be the nearest?

The output, for $n = 10$:

• left 0.28500 (too small by 0.048),
• right 0.38500 (too big by 0.052),
• middle 0.33250 (error $-0.0008$),
• trapezoid 0.33500 (error $+0.0017$).

The left and right sums are off by about 5 percent. The middle and trapezoid sums are about 30 times better with the same ten pieces. The last line shows that np.trapezoid(values, edges) gives the same as our trapezoid sum. That function adds the trapezoids for you.

How fast does the error fall?

Predict first. When the number of pieces doubles, by how much will the error change?

(The left and right rectangle sums are only first order: their error is halved when $n$ doubles. That is why trapezoids are the better first choice.)

With 7 pieces the error is already below 0.04. The code below lets you try 100 or 1000 pieces.

Data at uneven times

The trapezoid rule has a big advantage: it needs no formula, only the pairs of values and positions. And the pieces need not be equal.

Notice how this is the reverse of the last lesson. The slope of the position graph is the speed. The area under the speed graph is the distance. Differentiation and integration undo each other.

Common mistakes

• Giving np.trapezoid only the values. Give the x values too, unless they are one apart.
• Using the left rectangles for a rising curve and trusting the answer. It is too small.
• Mixing up the number of pieces and the number of points. $n$ pieces have $n + 1$ points.
• Forgetting the sign. An area below the axis counts as negative.
• Using np.trapz. In the new NumPy, the function is called np.trapezoid.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Area under a curve. You write the trapezoid rule.
2. Build. Open the problem How many pieces? and the problem Distance from measured speeds.

Remember these

• An area under a curve is a sum of narrow pieces. Left and right rectangles have errors proportional to the width.
• Trapezoids and the middle rule are much better: the error is proportional to the width squared.
• np.trapezoid(values, x) works for uneven gaps. cumulative_trapezoid gives the running total.
• Distance is the area under the speed graph. Work is the area under the force graph.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A rising curve is cut into rectangles at the left ends. Is the area too big or too small?

2. The trapezoid error with 10 pieces is 0.01. What do you expect with 20?

3. What does cumulative_trapezoid give for a speed graph?

4. What is the area under a force graph?