/sci/ - Science & Math

[Anonymous No.4142762]

Quick math question sci.

One of the fundamental theorems of integrals is that S(b top, a bottom) f(x) = - S(a top, b bottom) f(x)

Why exactly is this? I understand why all area underneath the X axis is negative, but I don't understand why flipping the way you "read" it will change the sign.

Also, in integrals, do you always do the top number of the S first, then the bottom, or raither, do you always "read" from the greater value to the lesser, and do it in that ordeR?

Sorry if I'm not clear, my math teacher doesn't really explain things at all, we just do it. Let me know if I need to clarify anything

[Anonymous No.4142775]

>>4142762

Define: S(b top, a bottom) f(x) as F(b) - F(a).

F(b) - F(a)
=-[ -F(b) + F(a) ]
= -[ F(a) - F(b) ]
= - S(a top, b bottom) f(x)

It's algebraic manipulation at the heart of it.

Also,

it's called the Fundamental Theorem of Calculus.

Use latex. I didn't do it because I'm lazy.

[Anonymous No.4142779]

>>4142775
Ok, that makes a bit more sense. Just verifying, all integrals are done from the greater value to the lesser one, or right to left, correct?

[Anonymous No.4142789]

>Use latex. I didn't do it because I'm lazy.

[Anonymous No.4142801]

>>4142779

It's like the definition the other anon used:

S(b top, a bottom) f(x) = F(b) - F(a)

It doesn't matter if b > a or b = a or b < a.

[Anonymous No.4142809]

>>4142789
Chuckled a bit.

[Anonymous No.4142818]

In the definition of the definite integral, each term of the Riemann sum has a factor (x_final - x_initial) / n. If x_final < x_initial then by definition the definite integral involves repeatedly multiplying the function by negative increments. So the sign of the entire integral flips.

[Anonymous No.4142827]

>>4142801
Ok, I get the math behind it now, thanks.

I'm just trying to think of a more intuitive way to think of it. If we have an integral of S(pi top, 0 bottom), its going to be 2 right?

if we have an integral of S(0 top, pi bottom), its going to be -2.

The area is therefore negative because it goes to the left of the "origin" of the equation, which is the bottom variable of the integral, in this case pi?? And thus alternatively, the area is postivie in this integral because the area is above the x axis to the right of the "origin", or 0.

And of course, the origin from which we integrate, for lack of a better word, is the bottom variable in the integral sign.

>that feel when I'm top of my entire class in calc and still don't understand this.

education system, man.

[Anonymous No.4142905]

>>4142827

That sounds about right. Just avoid using the word origin. It has a strict meaning already, that is the point (x = 0, y = 0);

If you want to get a slighly less "good enough to get you through the test but not really having any mathematical meaning" explanation, just remember that the Riemann integral is intuitively the limit of the sum of areas of rectangles under the curve as their width approaches zero.

When you flip the points where you're integrating (bottom with top), you flip the signal of the calculation of the area for each rectangle (because that signal depends on a subtraction of two points in your partition of the interval where you're integrating and the order of those points has been switched), thus each positive area becomes negative and vice versa, flipping the signal of the sum as a whole:

(-s_1) + (-s_2) + ... + (-s_k) = -1*(s_1 + s_2 + ... + s_k) = -S