The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles. A =

a2linetagadaW

a2linetagadaW

Answered question

2020-12-09

The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles. A=limnRn=limn[f(x1)Δx+f(x2)Δx+...+f(xn)Δx] Use this definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=7xcos(7x),0xπ2

Answer & Explanation

Brighton

Brighton

Skilled2020-12-10Added 103 answers

Split the interval into nn-subintervals. This gives you Δx, specifically, x=2πn.
This creates the following subintervales of [0,2π]
[0,2π/n],[2π/n,22π/n][2π/n,22π/n],[22π/n,32π/n][22π/n,32π/n],...,[(n1)2π/n,n2π/n][(n1)2π/n,n2π/n]
From here, your sample points, the xi‘s, will be chosen from each of these subintervals. Typically, the left hand points of each subintervals are chosen.
So you may express the area under the curve as:
limn2πn[7×0cos(7×0)+72πncos(72πn)+...+7n2πncos(7n2πn)

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