Hurkyl said:Nope. But the idea of the proof can still be used -- you can derive a general formula for
[tex]\int_{f(x)}^{g(x)} h(x, t) \, dt[/tex]
with suitable conditions on the three functions.
Hurkyl said:Have you tried not guessing, and actually trying to work it out, starting with the very definition of derivative?
The fundamental theorem of calculus is a fundamental concept in calculus that relates the concepts of differentiation and integration. It states that the definite integral of a function can be evaluated by finding an antiderivative of the function and evaluating it at the upper and lower limits of integration.
The fundamental theorem of calculus has two parts. The first part states that if a function is continuous on a closed interval, then the definite integral of the function can be evaluated by finding an antiderivative of the function and evaluating it at the upper and lower limits of integration. The second part states that if a function is continuous on a closed interval and its derivative is also continuous, then the derivative of the definite integral of the function is equal to the original function.
The fundamental theorem of calculus is important because it provides a way to evaluate definite integrals without using Riemann sums, which can be time-consuming and difficult. It also connects the concepts of differentiation and integration, making it a powerful tool for solving various problems in physics, engineering, and economics.
An antiderivative is a function whose derivative is equal to a given function. It is also known as the indefinite integral of the given function. In other words, an antiderivative is the reverse process of differentiation.
No, the fundamental theorem of calculus can only be applied to continuous functions. This means that the function must not have any breaks, holes, or jumps on the interval of integration. Additionally, the function must have an antiderivative that is also continuous on the interval of integration.