Question

Prove that the difference between the integration of a curve at two different points is equal to the area enclosed by the x-axis and the curve between these two points.

Answer 1

Basically prove\[\int\limits_{x _{0}}^{x _{1}}f (x) = AreaUnderCurve\]

Answer 2

uhh. i cant help srry ;(

Answer 3

which course is this for? The above identity is more a definition, a mathematical notation, what you might be looking for is called "Riemann integrability" ?

Answer 4

@Spacelimbus The course is "intellectual curiosity" :) So I checked out Riemann's integral, and it seems to be what my question is about. Although, i'm asking >why< is that definition correct, i.e. why does it yield the area under the curve (loosely speaking)?

Answer 5

Well I can only walk you again through the definitions again, it all depends on how rigorous you want things to be defined. To answer your question you might want to study the following expression:
\[\large R_n:= \sum_{k=1}^n \Delta x_k f( \xi_k) \]
If \(f: [a,b] \to \mathbb{R}\) is a conitnuous function you can always subdivide the interval \( [a,b]\) into \(n \) subintervals \([x_{i-1},x_i]\). You want to choose that interval so the following holds true:
\[\large a=x_0 < x_1 < \dots < x_n = b \]
now you're almost there. The length of every subinterval is given by \(\Delta x_i=x_i-x_{i-1}\). You want to make those intervals equal for each subinterval, so choose the maximum and then select a point in it \(\xi_i \in [x_{i-1},x_i]\).

Then the above definition \(R_n\) would be nothing but the summation of rectangles with dimension length times height given by \(\Delta x_k=\text{length}\) and \(f ( \xi_k)= \text{height}\).

Now if you sum up all this rectangles and it has to be given that \(\lim_{n \to \infty} \Delta_n =0 \) you call your expression Riemann integrable and write the Integral notation.

Answer 6

Ahhh thanks! That definition is pretty clear. I couldn't go through Wikipedia's because it looked sorta difficult, but this is all right. As a plus, I found this online: http://math.stackexchange.com/questions/15294/why-is-the-area-under-a-curve-the-integral the second answer is pretty intuitive, although it's not a rigorous proof.

Answer 7

Yes very good, math.stackexchange is one of the best online resources when it comes to mathematics :-) So you will find a lot of great quality answers there.