How Do You Picture Integrals
Favorites|Homepage
Subscriptions | sitemap
HOME > Mathematics > How Do You Picture Integrals

How Do You Picture Integrals

[From: ] [author: ] [Date: 11-05-16] [Hit: ]
Suppose that F is an anti-derivative of f on the interval [a, b]. If you chop up [a,b] into little pieces (form a partition),and look at one little sub-interval such as [x_0, x_1],......
I'm still having a hard time imagining how the area under a function relates to the definite integral.
I can derive the fundamental formula, and know that the definite integral is to take the limit of the riemann sum, as n approaches infinity.

For example, let say f(t) = t^2
∫ 3 0 f(x)dx = lim n→∞ ∑ n k=1 f(xk)Δx
= lim n→∞ ∑ n k=1 (3k/n)^2 * (3/n)
= lim n→∞ 27/n^3 * 1/6 * n(n+1)(2n+1)
= lim n→∞ 9/2n^2 * (2n^2 + 3n + 1)
= lim n→∞ 9 + 27/2n + 9/2n^2
= 9 which is x^3/3 when x = 3

I guess the question is how the riemann sum somehow transforms into the antiderivative of the function?

-
Yeah, the fundamental theorem of calculus is not intuitive----it's like magic! I'm not kidding, how one gets from a geometric interpretation like area under a curve to finding and evaluating an anti-derivative seems too good to be true!

It makes some intuitive sense when thinking about distance and velocity for example. If you chop up a rate of change curve (velocity) into a bunch of tiny little pieces and treat the curve like a sequence of constants over very small intervals, then total distance should be the sum of all products rate x times (integral of df/dt times dt).

That this generalizes is rather fantastic. The key to the proof is the Mean Value Theorem. Suppose that F is an anti-derivative of f on the interval [a, b]. If you chop up [a,b] into little pieces (form a partition),

a = x_0 < x_1 < x_2 < ...< x_n = b,

and look at one little sub-interval such as [x_0, x_1], the mean value theorem says that

F(x_1) - F(x_0) = F '(c)(x_1 - x_0)

for some c between x_0 and x_1. This means that

F(x_i) - F(x_(i-1)) = f(c_i) Δx_i

for each sub-interval in your partition---note that I used here that F '(c) = f(c). Add these up. On the left, you just get F(b) - F(a). On the right, you get ∫ f(x) dx over [a, b].

The Mean Value Theorem is the glue between differentiation and integration.

There are other fantastic results in calculus---all the fundamental theorems are like this in my opinion. Take Complex Variables when you get the chance. Residue calculus is equally thrilling!
1
keywords: Integrals,How,You,Picture,Do,How Do You Picture Integrals
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .