Why do we end up with f(x) instead of f(t) on the second fundamental theorem of calculus?
You are integrating f(t) between the limits a and x
f(a) is just a constant, and goes away when taking F'
but F(x) is a function and its derivative F'(x) is f(x).
t is just a dummy variable used to define the integrand.
The second fundamental theorem of calculus involves the derivative of an antiderivative, and it's important to understand why we typically use "f(x)" instead of "f(t)" in this theorem.
To explain this, we need to consider two functions: the original function f(x) and its antiderivative F(x). The independent variable, denoted as "x", represents the variable of integration or differentiation.
Now, the second fundamental theorem of calculus states that if we have a continuous function f(x) and find its antiderivative F(x), then the definite integral of f(x) from a to b is equal to F(b) minus F(a), where a and b represent the limits of integration.
The reason we use "f(x)" instead of "f(t)" in this theorem is that "x" is the variable of integration or differentiation. In the context of the theorem, it represents the variable over which we are integrating. This variable "x" is used consistently throughout the integral and its limits.
On the other hand, "t" is commonly used as a parameter in other contexts, such as when dealing with parameterized curves or when solving differential equations. In those cases, "t" represents a parameter along the curve or the independent variable of the differential equation solution.
So, to summarize, in the second fundamental theorem of calculus, we use "f(x)" because "x" represents the variable of integration or differentiation. By using "f(x)", we can clearly express the connection between the integrand and its antiderivative, providing a fundamental relationship in calculus.