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0:37 6:16 Suggested clip Calculating a Definite Integral Using Riemann Sums - Part 1 YouTubeStart of suggested clipEnd of suggested clip Calculating a Definite Integral Using Riemann Sums - Part 1
A Riemann Sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. We saw that as we increased the number of intervals (and decreased the width of the rectangles) the sum of the areas of the rectangles approached the area under the curve.
The midpoint Riemann sums is an attempt to balance these two extremes, so generally it is more accurate. The Mean Value Theorem for Integrals guarantees (for appropriate functions f) that a point c exists in [a,b] such that the area under the curve is equal to the area f(c)(ba).
If the graph is increasing on the interval, then the left-sum is an underestimate of the actual value and the right-sum is an overestimate. If the curve is decreasing then the right-sums are underestimates and the left-sums are overestimates.
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution.
The advantage of the midpoint method is that one obtains the same elasticity between two price points whether there is a price increase or decrease. This is because the formula uses the same base for both cases.
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