# Fundamental Theorem of Calculus

1. Jan 23, 2005

Question :$$\int_0^{49pi^2} (sin(sqrt(x))/(sqrt(x)) dx$$

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?

2. Jan 23, 2005

### dextercioby

The Fundamental Theorem of Calculus allows u to compute definite integrals of functions,using their antiderivatives.That's all to it...

$$\int_{0}^{49\pi^{2}} \frac{\sin\sqrt{x}}{\sqrt{x}} dx$$

??

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

Daniel.

3. Jan 23, 2005

### Kane O'Donnell

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. Jan 23, 2005

### HallsofIvy

Staff Emeritus
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: Jan 23, 2005