Using Scipy Multiple integration with a single variable

Question

I have a function for acceleration a(t) which I know if I integrate twice with respect to t, I can find position x(t). I am trying to find position at t = 10 seconds.

Since a(t) is not multivariable I am having trouble using the Scipy dblquad function to calculate the double integral I need. Please see what I have so far:

  def a(t):
      return (2.5 / (1 + math.exp((t-8)/0.8)))

  def upperbound():
       return 10

  def lowerbound():
      return 0

  x = dblquad(a,0,10,lowerbound,upperbound)

This does not work as from what I can gather dblquad needs a multivariabled a(t). Can anyone help?

Please find documentation here: docs.scipy.org/doc/scipy-0.19.1/reference/generated/… — Will C
– Will C, Commented Oct 25, 2017 at 19:29
Explain how your function is a A Python function or method of at least two variables: — hpaulj
– hpaulj, Commented Oct 25, 2017 at 20:52

Chris Mueller · Accepted Answer · 2017-10-26 13:38:09Z

4

You can use scipy's single variable integration twice to accomplish this

import math
from scipy.integrate import quad

def a(t):
    return (2.5 / (1 + math.exp((t-8)/0.8)))

lb, ub = 0, 10

integral = quad(lambda t: quad(a, 0, t)[0], lb, ub)[0]
print(integral)
# 86.28470375472537

This is necessary because what you want to achieve is not to integrate a function of two variables but rather to integrate a function of one variable twice. In the integral statement above, the inner quad integrates the function once but maintains the integral as a function of t. The outer quad integrates the function the second time over the defined limits.

Note that it is necessary to take the first argument from the output of quad because it outputs a tuple. The second argument is an upper bound on the error of the numerical integration.

edited Oct 26, 2017 at 13:38

answered Oct 25, 2017 at 19:50

Chris Mueller

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3 Comments

user2882635 Over a year ago

Hi Chris. I realize this question is a few yers old, but I'm trying to perform a double integration on a function with a single variable and then plot the function and the dual integrated function. Could you please explain why did you use square brackets and why did you define the a(t) function as a lambda in you answer above in the integral =... line

hidde-jan Over a year ago

The square brackets are because of the last sentence of the answer. quad returns a tuple and you only need the first element. The lambda is to create a function of one variable (t) that returns the integral of the function a from 0 to t, which can again be integrated by quad. The function in t represents an indefinite integral of a. The a function can also be a normal function. No difference with a lambda.

Chris Mueller Over a year ago

@user2882635 To reiterate the previous comment. Using lambda is simply a quick way to define a function. It could have been written separately as def first_integral(t); return quad(a, 0, t)[0]. The [0] after each call to quad is used because quad returns a lot of extra information about the integration which is not needed in this case. The first returned argument is the value of the integral and is all that is needed.

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