Second Fundamental Theorem of Calculus The Second Fundamental Theorem of Calculus extends the Fundamental Theorem of Calculus to Riemann integratio...
The Second Fundamental Theorem of Calculus extends the Fundamental Theorem of Calculus to Riemann integration. This theorem gives a powerful formula for approximating the definite integral of a function on a given interval using a Riemann sum.
Formally, the Second Fundamental Theorem of Calculus states the following:
If a function f(x) is differentiable on the interval [a, b], and its derivative is continuous on [a, b], then the definite integral of f(x) from a to b can be approximated by the Riemann sum:
∫_a^b f(x) dx ≈ (b-a) * f'(c)
where c is a point in (a, b).
Intuitively, this theorem says that the Riemann sum is an approximation of the definite integral by choosing the point c in the interval (a, b) based on the function's slope. The error of this approximation (the difference between the true value and the Riemann sum) decreases as the number of points used in the Riemann sum increases.
Key points:
The Second Fundamental Theorem of Calculus builds upon the Fundamental Theorem of Calculus by introducing the concept of Riemann sums.
It provides a rigorous way to approximate definite integrals using Riemann sums.
This theorem is applicable to functions whose derivatives are continuous on the interval [a, b].
Examples:
If f(x) = x^2, then f'(x) = 2x, which is continuous on [0, 1].
The definite integral of f(x) from 0 to 1 is 1/3, which can be approximated using the Riemann sum with n points as follows:
∫_0^1 f(x) dx ≈ (1-0) * (2^n)/(n!)
This approximation approaches 1/3 as n increases, demonstrating the accuracy of the Riemann sum