We’ve seen that the definite integral

\[\int_a^c f(x)dx\]

can be regarded as the area under the graph of \(y = f(x)\) over the interval [a, c]. If you have a point \(b\) such that \(a < b < c\), the above integral can be expressed as the sum of the integral from \(a\) to \(b\), and the integral from \(b\) to \(c\). This is the additive property of definite integrals.

\[\int_a^b f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx\]

To review a complete list of properties of integrals, check out this article.

The Area of a Region Bounded by Two Graphs

If you have a couple of functions \(f\) and \(g\) such that \(f(x) \geq g(x)\) over an interval [a, b], then the area of the region between the two curves is the difference of their definite integrals.

\[\int_a^b [f(x) - g(x)]dx\]

As an example, let’s find the area bounded in the following region.

import matplotlib.pyplot as plt
import sympy as sp
from sympy.abc import x
import numpy as np

f = sp.lambdify(x, 2*x + 1, "numpy")
g = sp.lambdify(x, x**2 + 1, "numpy")

xs = np.arange(-1, 5, 1)

plt.plot(xs, f(xs), color = "red", label="$f(x) = 2x + 1$")
plt.plot(xs, g(xs), label="$g(x) = x^2 + 1$")

plt.fill_between(xs, f(xs), g(xs), where=f(xs) >= g(xs), color = "yellow")
plt.legend(frameon=False)

sp.integrate(f(x), (x, 0, 2)) - sp.integrate(g(x), (x, 0, 2))
## 4/3

Life Science Example

A clever college student develops an engine that is believed to meet all state standards for emission control. The new engine’s rate of emission is given by

\[E(t) = 2t^2\] where \(E(t)\) is the emissions, in billions of pollution particulates per year, at time \(t\), in years. The emission rate of a conventional engine is given by

\[C(t) = 9 + t^2\]

  • At what point in time (how many years) will the emission rates be the same?
  • What reduction in emissions results from using the student’s engine?
from sympy.abc import t

E = 2*t**2
C = 9 + t**2

sp.solve(sp.Eq(E - C, 0), t)[1]
## 3
C = sp.lambdify(x, C, "numpy")
E = sp.lambdify(x, E, "numpy")

sp.integrate(C(t), (t, 0, 3)) - sp.integrate(E(t), (t, 0, 3))
## 18

Average Value of a Continuous Function

Another important use of the area under a curve is finding the average value of a continuous function over a closed interval.

Formally, let \(f\) be a continuous function over a closed interval [a, b]. It’s average value, \(y_{av}\), over the interval is given by

\[y_{av} = \frac{1}{b - a} \int_a^b f(x)dx\]

Let’s look an example. Let’s find the average of \(f(x) = x^2\) over the interval [0, 2].

xs = np.linspace(0, 2)

f = sp.lambdify(x, x**2, "numpy")

def avg(f, a, b):
  integral = sp.integrate(f(x), (x, a, b))
  return float(integral * (1 / b - a))
  
average = avg(f, 0, 2)
average
## 1.3333333333333333
plt.plot(xs, f(xs))
plt.fill_between(xs, f(xs), color="green", alpha=0.3)
plt.axhline(average, ls="--", color="gray")
plt.fill_between(xs, average, color="gray", alpha=0.3)