Evaluate integral of unknown function with data table

Question

I want to find the 2 bounded areas. However, i dont know the function of the blue curve but i do have a data table of it. I looked up and found out the possibility of using scipy.integrate.simps but i dont know how to specify that the area is bounded by the red line(either upper or lower) but not by the x-axis.

import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
import matplotlib
from matplotlib.ticker import (MultipleLocator, FormatStrFormatter,
                               AutoMinorLocator)
from sklearn.linear_model import LinearRegression
from scipy import interpolate
from scipy import integrate

%matplotlib inline

file = pd.read_excel("T8.xlsx","Phthalansäureanhydrid",usecols=[2,3])
X = file['Zeit(s)']
Y = file['Temperatur(Celcius Grad)']

fig, ax = plt.subplots()
ax.plot(X,Y,'-',color='#10A5F3', label="Phthalansäureanhydrid")
ax.grid(True, which='major', axis='both', color='#F19211', linestyle='-')
#ax.grid(True, which='minor', axis='both', color='#F19211', linestyle='--')
ax.spines['left'].set_position('zero')
ax.spines['right'].set_color('none')
#ax.spines['bottom'].set_position('zero')
ax.spines['top'].set_color('none')
#ax.legend(loc='upper center', frameon=True)

#major & minor ticks
ax.xaxis.set_major_locator(MultipleLocator(100))
ax.xaxis.set_major_formatter(FormatStrFormatter('%d'))
ax.xaxis.set_minor_locator(MultipleLocator(10))

#extrapolation - first line
temp1 = []
time1 = []
xnew1 = []
for i in file.index:
    if i > 630:
        temp1.append(file['Temperatur(Celcius Grad)'][i])
        time1.append(file['Zeit(s)'][i])
    else:
        xnew1.append(file['Zeit(s)'][i])

extrapo1 = InterpolatedUnivariateSpline(time1, temp1, k=1)
ynew1 = extrapo1(xnew1)

#extrapolation - second line
temp2 = []
time2 = []
xnew2 = []
for i in file.index:
    if 100<i<400:
        temp2.append(file['Temperatur(Celcius Grad)'][i])
        time2.append(file['Zeit(s)'][i])
    if i>400:
        xnew2.append(file['Zeit(s)'][i])

ynew2 = np.polyval(z,xnew2)
z = np.polyfit(time2,temp2,1)

#horizontal line
def hori(x):
    a = xnew1.index(x)
    b = xnew2.index(x)
    return np.linspace(ynew2[b],ynew1[a])

#integral

plt.plot(xnew1,ynew1,'-',color='black')
plt.plot(xnew2,ynew2,'-',color='black')
plt.plot([470]*len(hori(470)),hori(470),'--',color='red')

plt.savefig('phth.pdf')

Link to data: https://docs.google.com/spreadsheets/d/1xznXj-aA-Szq2s4KWb-qPWYxZbQNrA5FgUCQT6i7oVo/edit?usp=sharing

Answer 1

I'll code up a basic illustrative example so you can see how you might go about something like this. This is not for your data, but should help you along:

import matplotlib.pyplot as plt
import numpy as np

# Here are my curves
x = np.linspace(0, np.pi/2, num=100)
blue = np.sin(x)
orange = np.linspace(0.5, 0.8, num=100)
green = np.linspace(0.1, 0.2, num=100)

# Here is a plot to illustrate my example
plt.plot(x, blue, x, orange, x, green)
plt.show()

This plot has been annotated with the area that I want to work out; the area under the Blue curve but bounded by the Green and Orange curves:

So to work out this area we're going to work out the areas:

• Under the Blue curve when it's above Green and below Orange

  and add it to the area

• Under the Orange curve when it's below Blue

  then subtract the area

• Under the Green curve when it's below Blue

That's a little complex but try to picture those areas so that you can work out why we're adding and subtracting those areas. We can work out those areas using the np.trapz function and by masking only the areas that we want:

mask = (blue > green) * (blue < orange)
first_bullet_area = np.trapz(blue[mask], x[mask])
mask = (orange < blue)
second_bullet_area = np.trapz(orange[mask], x[mask])
mask = (green < blue)
third_bullet_area = np.trapz(green[mask], x[mask])

Then we just need to do that addition and subtraction:

>>> first_bullet_area + second_bullet_area - third_bullet_area
0.6190953349008973