dextercioby said:He cannot find the antiderivative among elementary functions...Yet he can solve the exercise without knowing it.
Daniel.
ProBasket said:isnt the "Fundamental Theorem of Calculus" just solving it as a regular integral? that's what i thought it was.
Hurkyl said:By the way, please tell me that you copied the problem down incorrectly and it actually says:
[tex]\int_x^{3} \sin t^3 \, dt[/tex]
If not, then bear in mind that your source is using poor notation -- they used the symbol x to represent two very different things.
The Fundamental Theorem of Calculus is a fundamental principle in calculus that relates the concepts of differentiation and integration. It states that the integral of a function f(x) over an interval [a,b] is equal to the difference of the antiderivative of f(x) evaluated at the endpoints a and b, also known as the definite integral. In simpler terms, it allows us to evaluate integrals by finding the antiderivative of a function and plugging in the endpoints of the interval.
The Fundamental Theorem of Calculus is essentially the bridge between derivatives and integrals. The first part of the theorem, called the First Fundamental Theorem of Calculus, states that differentiation and integration are inverse operations of each other. In other words, the derivative of the integral of a function is equal to the original function. The second part, called the Second Fundamental Theorem of Calculus, provides a method for evaluating definite integrals using antiderivatives.
The Fundamental Theorem of Calculus is important because it provides a powerful tool for solving problems in calculus. It allows us to evaluate integrals without having to use the limit definition of integration, which can be tedious and time-consuming. It also allows us to easily find the area under a curve, which has many real-world applications in fields such as physics, engineering, and economics.
No, there are two versions of the Fundamental Theorem of Calculus. The first version, as mentioned before, relates differentiation and integration as inverse operations. The second version provides a method for evaluating definite integrals using antiderivatives. Both versions are equally important and are often used together in solving problems in calculus.
Yes, there are some limitations to the Fundamental Theorem of Calculus. It only applies to continuous functions and only works for definite integrals. It also assumes that the function being integrated is integrable, meaning it must have a finite number of discontinuities. Additionally, it cannot be used to evaluate improper integrals, which have infinite bounds or integrands that tend to infinity.