Course 2.2Lesson 3 of 5

Integrals with SciPy

scipy.integrate.quad integrates a function of one variable between two limits and returns the value and an estimate of its error, usually to about 10 digits, by choosing its own pieces. It works for functions that have no formula for their antiderivative, such as e to the minus x squared, and for infinite limits written with np.inf, so the integral of e to the minus x squared over the whole line is the square root of pi. The work of a variable force is the integral of the force over the distance, for a spring from a to b it is one half k times b squared minus a squared, and for a stiffer spring the same call gives the work at once. quad takes a function of one number, and the answer should be checked against a case with a known value.

about 35 min1 simulation2 practice questions

Lesson 3 of 5Show all lessons

Try the code and the simulations beside the lesson.

What you will be able to do

After this lesson you can

integrate a function with scipy.integrate.quad,
integrate functions that have no formula for their antiderivative, and with infinite limits, and
find the work done by a variable force.

One call for an integral

With trapezoids, you choose the number of pieces, and the answer is as good as the number. SciPy has a method that chooses its own pieces, finer where the function changes quickly and coarser where it is smooth, until the answer is accurate to about ten digits.

Predict first. What is the area under $\sin x$ from 0 to $\pi$ ? Then run it.

quad(f, a, b) integrates the function f from a to b. It gives two numbers: the value and an estimate of the error. The error estimate is tiny here, below $10^{-10}$ , which is shown by the True.
The function f must take one number and give one number. NumPy functions such as np.sin are fine. A lambda is the easy way for your own formulas: lambda x: x ** 2.
The first line is the known answer 2. The second is $\int_0^1 x^2\,dx = \tfrac{1}{3}$ , and we printed the exact third for comparison.
The third integral is $\int_0^1 e^{-x^2}\,dx$ . There is no formula for the antiderivative of $e^{-x^2}$ , so no amount of algebra gives the answer. quad finds it at once: 0.746824. This is the kind of integral for which numerical methods are indispensable.

Infinite limits

quad can fail if the integral does not exist. For $\int_1^\infty \dfrac{1}{x}\,dx$ , the area is infinite, and quad gives a warning and a wrong number. If a result looks strange, think about whether the integral exists.

Work done by a changing force

When a force $F$ is constant, the work over a distance $d$ is $Fd$ . When the force changes with the position, the work is the integral of the force over the distance: the area under the force graph.

Predict first. A spring with $k = 50$ N/m is stretched from 5 cm to 15 cm. The force goes from 2.5 N to 7.5 N. What is the work, roughly?

Run it.

- The force of a spring is $F = kx$ . The work to stretch it from $a$ to $b$ is $\int_a^b kx\,dx = \tfrac{1}{2}k(b^2 - a^2)$ . With the numbers, $\tfrac{1}{2} \times 50 \times (0.0225 - 0.0025) = 0.5$ J. The integral and the formula agree. A rough check: the average force, 5 N, times the distance, 0.1 m, is 0.5 J.

- A stiffer spring has the force $F = 50x + 200x^3$ (a real spring gets stiffer when it is stretched far). For this spring, a formula for the work needs one more step of calculus. quad gives the answer without effort: 1.08 J from 0 to 0.2 m. (By hand: $25 \times 0.04 + 50 \times 0.0016 = 1.0 + 0.08$ .)

The method does not care how hard the formula is.

The red box of height 1 and width 1/k has exactly the area under the whole curve: what the tail adds far out is what the box adds near the start.

Choosing a method

If you have a formula, use quad.
If you have measured values, use np.trapezoid(values, x).
For a check, find a case with a known answer (a constant function, or $\sin$ over a full half-wave), and test your code on it first.

Common mistakes

Forgetting that quad gives two numbers. value, error = quad(...).
A function that takes an array. quad calls the function with one number at a time.
Giving quad a function that needs a list. It calls the function with one number at a time.
Not checking the result with a known case.
Integrating a function with a break or a pole between the limits.

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:

Warm-up. Open the lab Integrals with quad. You write functions that use quad and check them.
Build. Open the problem The bell curve and the problem Work of a stiff spring.

Remember these

quad(f, a, b) gives the value and an error estimate. It chooses the pieces by itself.
It works for functions with no antiderivative and for infinite limits (np.inf).
Work is the integral of force over distance.
Use quad for a formula and np.trapezoid for measured data. Always test on a known case.