Fundamental Theorem of Calculus – Part 1 and 2

Introduction to Fundamental Theorem of Calculus

Fundamental Theorem of Calculus – Part 1 and 2: The fundamental theorem of calculus is a theorem in mathematics that states that the integral of a function is the derivative of the function evaluated at the interval over which the integral is taken. In symbols,

The fundamental theorem of calculus is one of the most important results in calculus. It allows us to compute derivatives and integrals using each other’s properties.

Importance of Fundamental Theorem of Calculus in Mathematics

The fundamental theorem of calculus is a theorem in mathematics that states that differentiation and integration are inverse operations. This theorem is important because it allows mathematicians to find derivatives and integrals of functions using simple algebraic methods.

Fundamental Theorem of Calculus: Integrals & Anti Derivatives

However the Fundamental Theorem of Calculus is a theorem in calculus that states that the integral of a function is equal to the derivative of the function evaluated at the point where the integral is taken. In symbols,

[\int_a^b f(x)dx = f(b) – f(a).\]

The theorem first discussed by Isaac Barrow in his 1670 book Lectiones geometrical The first rigorous proof given by Gottfried Wilhelm Leibniz in 1675.

Fundamental Theorem of Calculus Part 1

If a function defined on an interval and is a number in that interval, then there exists a number such that

provided that is continuous on the interval .

Fundamental Theorem of Calculus Part 2

If is a continuous function on the interval , then the derivative of is

provided that is differentiable on .

Practice Problem

A thin metal plate subjected to a tensile load. The plate has a width of 1.00 cm, a thickness of 0.01 cm, and a length of 10.0 cm. The applied load is 1.00 N.

What is the stress in the plate?

The stress in the plate is 1.00 N/0.01 cm = 100.0 N/cm.

Second Fundamental Theorem of Calculus

Therefore the Second Fundamental Theorem of Calculus states that the derivative of the integral of a function is the function itself.