Fundamental Theorem of Calculus

[ProBasket]

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?

 

[dextercioby]

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

Is your integral
\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.

 

[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

 

[HallsofIvy]

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".