Fundamental Theorems of Calculus
Overview
In simple terms these are the fundamental theorems of calculus:
When we know the indefinite integral:
We can then calculate a definite integral between a and b by the difference between the values of the indefinite integrals at b and a:
Let's explore the details:
First Fundamental Theorem of Calculus
For a continuous function f(x) on an interval [a, b] with the integral:
then the derivative of the integral F(x) gets us the original function f(x) back again:
This means that the derivative of the integral of f with respect to its upper limit is the function f itself.
Example: f(x) = 2x
The integral of 2x is x2, and using the second theorem (below):
Taking the derivative:
So the derivative of the integral of 2x got us 2x back again.
Example: Constant speed
A car travels at a constant speed of 50 km per hour for exactly one hour:
Note: "C" is however far the car had traveled already.
Second Fundamental Theorem of Calculus
When we have a continuous function f(x) on an interval [a, b], and its indefinite integral is F(x), then:
In other words the definite integral of f(x) from a to b equals the difference in the values of F(x) at b and a
This makes calculating a definite integral easy if we can find its indefinite integral.
Example
Imagine we're filling a tank with water, and the rate at which water flows into the tank is given by f(t), where t is time in minutes.
If F(t) measures the total volume of water in the tank at any time t, then the amount of water added to the tank between times a and b is F(b) - F(a).