The fundamental theorem of calculus establishes a connection between the definite integral and the indefinite integral. It states that, for any continuous funct...
The fundamental theorem of calculus establishes a connection between the definite integral and the indefinite integral. It states that, for any continuous function f(x) defined on a closed interval [a, b], the definite integral from a to b, denoted by ∫a^b f(x) dx, is equal to the indefinite integral from a to b, denoted by ∫_a^b f(x) dx.
In other words, the definite integral represents the area under the curve f(x) between the limits a and b, while the indefinite integral represents the area under the curve f(x) from a to b. The fundamental theorem establishes that these two integrals are equal, meaning they represent the same numerical value.
For example, consider the function f(x) = x^2. The definite integral from 0 to 1 of f(x) dx is equal to the indefinite integral from 0 to 1 of x^2 dx. Both integrals evaluate to 1/3, which is the area under the curve f(x) = x^2 between 0 and 1.
The fundamental theorem of calculus has numerous applications in economics and other fields. It allows us to evaluate definite integrals, which are used to find areas, volumes, and other properties of curves. It can also be used to find the derivative and integral of functions, which are fundamental concepts in economics and finance