Areas/Volumes in Integration

rectangles man

or fundamental theorem of calculus?

what is areas and volumes

https://proofwiki.org/wiki/Fubini%27s_Theorem

>>8657890
The solid proof is that as the rectangles that make up the upper sum get smaller and smaller their area converges to the same value as the rectangles that make up the lower sum. Since the lower sum is strictly less than the area/volume and the upper sum is strictly more than the area/volume/whatever then it follows that the area is that what you're looking for.

>>8657901
>>8658466
Stupid Americans. Learn Analysis.

>>8658470
>when maths became the study of calling different things by the same name, one brave anon decided to call the same thing different names

>>8657890
Daily reminder that an integral only corresponds to area in very specific cases.

>>8658502
no

>>8659216
Even in OP's picture, as part of the cure is below the x-axis, if you were to integrate you would get a number smaller than the actual area.

>>8658502
daily reminder that using integration to find the area is reserved for intro calc classes as it's a convenient example for practicing integrals but nobody has ever once calculated an area integral once past calc II so nobody cares that integrals only correspond to areas in specific cases

>>8659223
No
If you insist, let's add 1 gorillion to both f(x) and g(x), creating F(x) and G(x) respectively. Now we have satisfied your stupid requirement that the functions have to be positive.
We calculate the area by integrating the difference between values of F and G, getting integrate from x1 to x2 (F(x)-G(x))) with respect to x.

Let's not beat around the bush, F(x) - G(x) = f(x)+gorillion-(g(x)+gorillion) = f(x)-g(x) = anon is a faggot

>>8658502
What do you mean exactly?

Maybe you mean that there are several types of mathematical objects called "integral", such as the Riemann integral or the Lebesgue integral. Do you mean that sometimes there are definitions that don't simply interpret the integral as an infinite sum of areas?

>>8659216
Take for example the expected value of a random variable. It's a Lebesgue integral over the sample space.

>>8657890
Integration is defined as the area between the curve and the x-axis. Negative areas are permitted under the definition for curves with negative values. The fundamental theorem of calculus relates the integral to the derivative. The FTC can be easily generalized and extended to n-dimensional "volumes".

>>8659223
Area between curves isn't dependant on the x axis

>>8658477
I'd just like to interject for a moment. What you're referring to as calculus, is in fact, real analysis, or as I've recently taken to calling it, [math]\left( \mathbf R,\, +,\, \times,\, \leqslant,\, \left| \cdot \right|,\, \tau \,=\, \left\{ A \,\subset\, \mathbf R \mid \forall x\,\in\, A,\, \exists \varepsilon \,>\, 0,\, \left] x \,-\, \varepsilon,\, x \,+\, \varepsilon\right[ \,\subset\, A \right\},\, \bigcap_{\begin{array}{c} A \,\sigma \text{-algebra of}\, \mathbf R \\ \tau \,\subset\, A \end{array}} A,\, \ell \right)[/math]-analysis. Calculus is not a branch of mathematics unto itself, but rather another application of a fully functioning analysis made useful by topology, measure theory and vital [math]\mathbf R[/math]-related properties comprising a full number field as defined by pure mathematics.

Many mathematics students and professors use applications of real analysis every day, without realizing it. Through a peculiar turn of events, the application of real analysis which is widely used today is often called "Calculus", and many of its users are not aware that it is merely a part of real analysis, developed by the Nicolas Bourbaki group.

There really is a calculus, and these people are using it, but it is just a part of the field they use. Calculus is the computation process: the set of rules and formulae that allow the mathematical mind to derive numerical formulae from other numerical formulae. The computation process is an essential part of a branch of mathematics, but useless by itself; it can only function in the context of a complete number field. Calculus is normally used in combination with the real number field, its topology and its measured space: the whole system is basically real numbers with analytical methods and properties added, or real analysis. All the so-called calculus problems are really problems of real analysis.

>>8657890
Integration is sort of defined to be what we mean by area.
Said another way, what do you mean by the area under of the curve if not the integral?

>>8660254
good pasta