Fundamental Theorem of Calculus

In summary, the conversation discusses the use of the Fundamental Theorem of Calculus to compute definite integrals of functions using their antiderivatives. It is named as such due to the historical and mathematical significance of linking integration and differentiation. The theorem allows for easier computation of integrals, which would otherwise be tedious to calculate from first principles.
  • #1
ProBasket
140
0
Question :[tex]\int_0^{49pi^2} (sin(sqrt(x))/(sqrt(x)) dx[/tex]


should i just solve it as a regular integral like usally and then do F(b) - F(a)? if so, why is it called Fundamental Theorem of Calculus if it's just like a regular integral?
 
Physics news on Phys.org
  • #2
The Fundamental Theorem of Calculus allows u to compute definite integrals of functions,using their antiderivatives.That's all to it...

Is your integral
[tex] \int_{0}^{49\pi^{2}} \frac{\sin\sqrt{x}}{\sqrt{x}} dx [/tex]

??

If so,then a simple substitution will allow u to use the theorem.

Daniel.
 
  • #3
The reason why it's called fundamental is mostly historical, but partly because it's quite remarkable that integration and differentiation are linked. On the face of it, the theory of derivatives is about limits of functions at a point, whereas integration is about the limits of sums, so it's not immediately obvious that the two are linked. The FTC shows that they are.

(pretty useful result to have - calculating integrals from first principles would be very tedious!)

Kane
 
  • #4
Why is WHAT called "Fundamental Theorem of Calculus"? Certainly it's not this integral which is what your question seems to imply!

It is the "Fundamental Theorem of Calculus" that allows you to do "regular integrals".
 
Last edited by a moderator:

1. What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a theorem in calculus that connects the two branches of calculus: differential calculus and integral calculus. It states that the derivative of a function is equal to the original function evaluated at the upper limit of integration, minus the original function evaluated at the lower limit of integration.

2. Why is the Fundamental Theorem of Calculus important?

The Fundamental Theorem of Calculus is important because it allows us to easily calculate the definite integral of a function by finding its antiderivative. This simplifies the process of computing areas under curves and solving many other problems in mathematics, physics, and engineering.

3. How is the Fundamental Theorem of Calculus used in real life?

The Fundamental Theorem of Calculus is used in various real-life applications, such as calculating the area under a curve to find the total distance traveled by an object, determining the velocity and acceleration of an object from its position function, and solving optimization problems in economics and engineering.

4. Is there more than one version of the Fundamental Theorem of Calculus?

Yes, there are two versions of the Fundamental Theorem of Calculus: the first version, which relates the derivative and integral of a function, and the second version, which relates the integral of a function to its antiderivative. Both versions are equally important and are used in different situations.

5. What are some common mistakes when applying the Fundamental Theorem of Calculus?

Some common mistakes when applying the Fundamental Theorem of Calculus include forgetting to use the chain rule when finding the derivative of a composite function, incorrectly evaluating the limits of integration, and mistakenly assuming that the antiderivative exists for a given function. It is important to carefully follow the steps and conditions outlined in the theorem to avoid these errors.

Similar threads

  • Introductory Physics Homework Help
Replies
2
Views
417
  • Calculus
Replies
6
Views
1K
Replies
2
Views
173
  • Introductory Physics Homework Help
Replies
10
Views
814
  • Calculus and Beyond Homework Help
Replies
15
Views
745
  • Introductory Physics Homework Help
Replies
23
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
500
  • Introductory Physics Homework Help
Replies
28
Views
270
  • Calculus and Beyond Homework Help
Replies
10
Views
289
  • Introductory Physics Homework Help
Replies
25
Views
936
Back
Top