How Does Summation Approximate Area Under a Curve as n Approaches Infinity?

[Firben]

Interpret the given sum Sn as a sum of areas of rectangles approximating the area of a certain region in the plane, and hence evaluate (x→ infinity) lim Snn
Sn ∑2/n(1-((2i)/n))
i=1

http://s716.photobucket.com/albums/ww168/Pitoraq/?action=view&current=Rms2.jpg
(number 17)

Attempt at solution:

I guess that the function is y = 1-2x is it right ?

How to solve for lim when x →infinity ?

 

[HallsofIvy]

Not quite. Notice that the $\Delta x$ is 2/n. And so "x" is 2i/n. Or, you could factor that 2 out of the sum so that $\Delta x$ is 1/n. In that case, "x" is indeed i/n. Do you see that those both give the same integral?

 

[Ray Vickson]

Interpret the given sum Sn as a sum of areas of rectangles approximating the area of a certain region in the plane, and hence evaluate (x→ infinity) lim Sn


n
Sn ∑2/n(1-((2i)/n))
i=1

http://s716.photobucket.com/albums/ww168/Pitoraq/?action=view&current=Rms2.jpg
(number 17)

Attempt at solution:

I guess that the function is y = 1-2x is it right ?

How to solve for lim when x →infinity ?

x does NOTgo to infinity; n does. You could use standard algebraic formulas to evaluate the sum explicitly as a function of n, then let n go to infinity in that formula. You should have seen already all the summations you need, somewhere in your course notes or your textbook.

RGV