What does a Riemann sum represent?

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A Riemann Sum starts after a while the inquiry of how to ascertain the area beneath a deflexion (i.e., between a settled deflexion and the x-axis, which is essentially a Geometry inquiry). It turns out, though, that as after a while divers seemingly scant inquirys in mathematics, we can amplify the collision of the retort to this inquiry to other areas (e.g., ascertaining space-between travelled by a tender scene in physics). In regulate to do this, we must amalgamate the moderate inquiry using what is named a Riemann Sum.

We start after a while ascertaining the area beneath a settled deflexion (i.e., a deflexion after a while settled y-values). For pattern, deem we endeavor to ascertain the area beneath the deflexion ##f(x) = x^2## from ##x = 1## to ##x = 3##. We allot this space-between into "subintervals." For pattern, we jurisdiction pick-out immodest subintervals of similar width (1-1.5, 1.5-2, 2-2.5, 2.5-3).

We then trench the area of each subspace-between using a rectangle. The width of the rectangle is the the alter in x which is 0.5 (or ##Delta x = 0.5##). The tallness is a chosen y-rate for each subinterval. In this circumstance, we shall pick-out the right-hand endpoints. Thus we own the forthcoming athwart areas:

##f(1.5)*0.5+f(2)*0.5+f(2.5)*0.5+f(3)*0.5 = 10.75##.

Note that we can diminish this using summation (sigma) notation as:

##sum_(i=1)^4(f(1+i*0.5)*(0.5))## or further unconcealedly as ##sum_(i=1)^4(f(1+i*Deltax)*(Deltax))##.

We now finally end to the purpose of a Riemann Sum. A Riemann Sum is encountered when we relent our rules for situations relish the anterior one to avow for a further unconcealed scene of the tenor. That is, we could use irrelative rates for each ##Delta x## and excellent several points for each ##f(x)##. We could too understand functions that are not frequently settled.

The Riemann Sum would then contemplate relish this: ##sum_(i=1)^n(f(barx_i)*(Deltax_i))##, where ##barx_i## is some x rate in the ith. subspace-between and ##Deltax_k## is the width of that subinterval.

Some of our products jurisdiction now be denying, if some of the y-values are denying. These would reproduce-exhibit "denying areas" (i.e., areas adown the x-axis). The Riemann Sum would be an vestibule of the variety between the "positive" and "negative" areas.


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