Please take a look at this small description calculus help

[khanna203]

i have to describe the construction of the Riemann Integral... in 4-6 sentences.. and i was wondering.. if this is right.. and explains what the question is asking

In order to understand how the Riemann Integral is, we have to understand how area under a curve is taken from a graph. Given n amount of rectangles, the approximated area would simply be Σf(xi)Δx, Δx being the widths and f(xi) being the heights, which is known as a Riemann sum. When you take the lim as n ---> ∞, you get infinitely small rectangles which give the exact area under the curve. Since the integral is defined as the area under the curve, we get the Riemann Integral.

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[estro]

I would explain more how we divide the segment into rectangles and why when their length approaches infinity the integral approaches the "area under the curve". [partial sums]

 

[LCKurtz]

Since the integral is defined as the area under the curve, we get the Riemann Integral.

The integral is not defined as the area under the curve. The integral is defined by the limit of the Riemann sums. Then the area under the curve, assuming f(x) ≥ 0, is defined as the value of the integral, or the limit of the Riemann sums, they being the same.

 

[HallsofIvy]

You should also note that the interval the $\Delta x$ does not have to be constant- that is the interval does NOT have to be divided into sub-intervals of equal length.