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Tyler

Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
Tyler
Is there a different way to find the sum of a series for a nonlinear series? I have a summation based equation that if done by hand, adding each term, evaluates to an approximation to the definite integral of a line. But if the sum of a finite series equation is used, it evaluates to the integral of a line from the f(lower limit) to the f(upper limit) of the integral.

 Description: The equation. Filesize: 1.06 KB Viewed: 2020 Time(s)

26 Sep 2010, 22:50
MHajduk

Joined: 30 Mar 2006
Posts: 6038
Location: Poland
MHajduk
Your formula is improper mainly because you have forgotten integrated function symbol and you've "lost" somewhere limes on the right side of the equation.

This should be more accurate:

Integral (in the simple words) is a limit of the sums of series for the n → ∞
27 Sep 2010, 07:54
Tyler

Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
Tyler
I'm new to calculus, and learning it independently. I wrongly thought that the "f(x)dx" was the formula to find the solution. What is "dx" anyway? Some call it a differential, is that the same as a derivative? As for the limits, it originally had "delta X -> 0". In theory, it adds the infinite rectangles in the integral, with f(n delta x) being the height, and delta x being the width.
27 Sep 2010, 22:01
MHajduk

Joined: 30 Mar 2006
Posts: 6038
Location: Poland
MHajduk
Tyler wrote:
What is "dx" anyway? Some call it a differential, is that the same as a derivative?
By 'dx', 'dy', 'dz', 'dt' etc. etc. we denote small change of the value of the variables 'x', 'y', 'z' and 't' respectively. Sometimes differential is called 'infinitesimally small change of the variable value'.

And yes, concepts of the differentials and derivatives are related: sometimes derivatives of the functions are denoted as a quotients of the differentials.

More detailed explanations of what I wrote above you may find here:
28 Sep 2010, 08:39
Tyler

Joined: 19 Nov 2009
Posts: 1216
Location: NC, USA
Tyler
So they are essentially the same thing? The real integral equation is just height, or f(x), times width, or the [infinitesimally small] change in x?

What would be the differential of x be, in the context of an xy graph? Doesn't it change constantly?
02 Oct 2010, 21:01
MHajduk

Joined: 30 Mar 2006
Posts: 6038
Location: Poland
MHajduk

Basically, definite integral of the function f (f: R → R, where R is a set of the real numbers) with the low limit a and high limit b is interpreted usually as a area under curve f(x) (exactly sum of the areas under the curve f(x) and over the x axis [for the parts where f(x) ≥ 0] and areas under the x axis and over the f(x) curve taken with minus sign [for the parts of the chart where f(x) < 0]) between the points x=a and x=b. Equation written by me in the one of the previous posts shows how we can calculate approximations of the integral value. With the n getting bigger (consequently Δx getting smaller) approximation is getting better.

k is a number of the vertical strip under the f(x) curve (from x=a+kΔx to x=a+(k+1)Δx),

f(a + kΔx) is a height of this strip,

Δx is a width of the each of the strips (here, for simplicity, we assumed that all strips have the same width).
02 Oct 2010, 22:39
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