计算曲线下面积时面积的负值

Question

I have obtained coefficients of a polynomial using polyval in python. I used the following code to calculate area of the polynomial. The curve appeared above y=0 line when visualised. But the area when calculated appeared to be negative. Could you please someone help me what is the issue with the following code.

import numpy as np
# insert the coefficents of the polynomial
p = np.array([ 2.60349395e+09, -3.34913329e+10,  1.87588633e+11, -5.85791667e+11,
           1.04784500e+12, -8.03652884e+11, -8.37573977e+11,  3.17205629e+12,
           -4.33851893e+12,  3.48269443e+12, -1.71966834e+12,  4.86703099e+11,
           -6.07938187e+10])

def f(x):
    return ((p[0]*(x**12))+(p[1]*(x**11))+(p[2]*(x**10)))+(p[3]*(x**9))+(p[4]*(x**8))+
            (p[5]*(x**7))+(p[6]*(x**6))+(p[7]*(x**5))+(p[8]*(x**4))+(p[9]*(x**3))+
            (p[10]*(x**2))+(p[11]*(x**1))+(p[12]))

N = 1000
#insert the initial value of independent variable
x1=1
#insert the final value of the independent variable
x2 = 1.578687

dx = (x2 - x1)/N

A = 0
t=x1

while t<=x2:
  dA=f(t)*dx
  A = A +dA
  t = t + dx

# Area under the curve
print("A=",A)

Answer 1

The area is negative because the function f(t) is strictly negative in that interval. You can see that by quickly plotting the function making use of np.arange and the matplotlib package.

import numpy as np
import matplotlib.pyplot as plt
# insert the coefficents of the polynomial
p = np.array([ 2.60349395e+09, -3.34913329e+10,  1.87588633e+11, -5.85791667e+11,
          1.04784500e+12, -8.03652884e+11, -8.37573977e+11,  3.17205629e+12,
          -4.33851893e+12,  3.48269443e+12, -1.71966834e+12,  4.86703099e+11,
          -6.07938187e+10])

def f(x):
   return (p[0]*(x**12) + p[1]*(x**11) + p[2]*(x**10)+ p[3]*(x**9) + p[4]*(x**8) +
           p[5]*(x**7) + p[6]*(x**6) + p[7]*(x**5) + p[8]*(x**4) + p[9]*(x**3) +
           p[10]*(x**2) + p[11]*x+p[12])

plt.plot(np.arange(0, 2, 0.001), f(np.arange(0, 2,  0.001))) # plotting

You can also use exceptions to check for negative values of f(t) when integrating:

N = 1000
#insert the initial value of independent variable
x1 = 1
#insert the final value of the independent variable
x2 = 1.578687

dx = (x2 - x1)/N

A = 0
t = x1

while t<=x2:
    if f(t) < 0:
        # raise exception if negative value is encountered, integration will stop.
        raise Exception("Negative values encountered") 
    else:
        dA = f(t)*dx
        A = A +dA
        t = t + dx
        


# Area under the curve
print("A=",A)

Output:

---------------------------------------------------------------------------
Exception                                 Traceback (most recent call last)
<ipython-input-38-2aea71dc1318> in <module>
     12 while t<=x2:
     13     if f(t) < 0:
---> 14         raise Exception("Negative values encountered")
     15     else:
     16         dA = f(t)*dx

Exception: Negative values encountered