Fundamental Theorem of Calculus.

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Find area under the graph f(x) =x between x=0 and x=1 by taking the limit of the sum of approximating rectangles whose heights are the values of the function at the right hand end point of each subinterval.
Solution. Let f(x)=x. Let us equally partition the interval [0, 1 into n subintervals: [0, 1/n], [1/n, 2/n], [2/n, 3/n], ..., [(n-1)/n, 1], and ,
Then we can take the right hand end point of each subinterval to form the following sum:

So, the area is equal to

Indeed, the area is equal to

by the Fundamental Theorem of Calculus.

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