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Eeh, Riemann sums give the rationale for many numerical integration schemes..HallsofIvy said:The only purposes of Riemann sums are to give a basic definition of the Riemann integral and, sometimes, to reason out how to set up an integral. You never use Riemann sums to actually do an integral. Find an anti-derivative, evaluate it at the upper and lower bounds, and subtract.
Yes, thank you. I didn't think of that. However, I would be inclined to say that most methods, such as Simpson's rule, can be interpreted as approximating the function, over sections of the interval rather than Riemann sums.arildno said:Eeh, Riemann sums give the rationale for many numerical integration schemes..
True enough.HallsofIvy said:However, I would be inclined to say that most methods, such as Simpson's rule, can be interpreted as approximating the function, over sections of the interval rather than Riemann sums.
arildno said:The force of Riemann sums, as I see it, is the ease by which proofs in full generality may be given by means of them.
What you are doing is integrating not differentiating.clm222 said:wait a second, using
[itex]\int_a^b f(x) dx = F(b)-F(a)[/itex]
I don't even need to know the function, just the values at point a and point b, am i correct?
One more question: given a function, say: f(x)=x2
to find the area under a curve (a to b) would I need to differentiate:
[itex]\int_a^b[/itex] 2x dx
After integrating, you would not still use the integral sign- that's been done. What you are doing is evaluating [itex]x^2[/itex].or would I simply plug in the function?
[itex]\int_a^b[/itex] x2 dx
I don't think anyone has answered this part.clm222 said:wait a second, using
[itex]\int_a^b f(x) dx = F(b)-F(a)[/itex]
I don't even need to know the function, just the values at point a and point b, am i correct?
clm222 said:Ok, and that is the area of the function?
Yeah, I confuse easily!Muphrid said:Careful, I think you confused HallsOfIvy.
A definite integral is a mathematical concept used to find the area under a curve between two points on a graph. It is represented by the symbol ∫ and has a lower and upper limit that define the points between which the area is being calculated.
To calculate a definite integral, you need to first find the anti-derivative of the function. Then, substitute the upper and lower limits into the anti-derivative and subtract the result of the lower limit from the upper limit.
A definite integral has specific limits and gives a numerical value, while an indefinite integral has no limits and gives a general function with a constant of integration.
Definite integrals are important in many fields of science and engineering, as they are used to calculate quantities such as area, volume, and displacement. They are also used in solving differential equations and modeling real-world phenomena.
Some common methods for calculating definite integrals include the Riemann sum, the trapezoidal rule, and Simpson's rule. These methods use different approximations to find the area under a curve and can be used for both analytical and numerical integration.