- #1
smize
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I know I should know this, but how would one convert a typical integral into a Rieman Sum?
∫0n sinx + x dx for whatever n.
for example.
∫0n sinx + x dx for whatever n.
for example.
smize said:I know I should know this, but how would one convert a typical integral into a Rieman Sum?
∫0n sinx + x dx for whatever n.
for example.
Well, since the function [itex]\,f(x)=\sin x + x\,[/itex] is continuous everywhere, it is Riemann integrable in any finitesmize said:I know I should know this, but how would one convert a typical integral into a Rieman Sum?
∫0n sinx + x dx for whatever n.
for example.
The purpose of converting an integral to a Riemann sum is to approximate the area under a curve using smaller, simpler shapes (usually rectangles). This allows for easier computation and a more accurate representation of the total area.
The steps to convert an integral to a Riemann sum are:
The difference between a left Riemann sum and a right Riemann sum lies in the placement of the rectangles. In a left Riemann sum, the left endpoint of each subinterval is used to determine the height of the rectangle. In a right Riemann sum, the right endpoint of each subinterval is used instead. This can result in slightly different approximations of the total area under the curve.
Taking a limit as the number of subintervals approaches infinity allows us to find the exact value of the integral. As the number of subintervals increases, the rectangles become smaller and the approximation of the total area becomes more accurate. Taking the limit ensures that we are considering all possible subintervals and their corresponding areas, resulting in the exact value of the integral.
Real-world applications of converting an integral to a Riemann sum include calculating the total distance traveled by an object with varying speed, determining the total volume of a three-dimensional shape with changing cross-sectional areas, and finding the total amount of work done by a force that varies over a given distance.