The area A of the region S that lies under the graph of the continuous function

Question

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 = lim_n rightarrow infinity R_n = lim_n rightarrow infinity[f(x_1)delta x + f(x_2)delta x + ... + f(x_n) delta x] 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) = 9x cos(9x), 0 lessthanorequalto x lessthanorequalto pi/2 A = lim_n rightarrow infinity sigma^n_i = 1 9 pi^2/4n^2 4n^2 cos (9(pi/2n))

Explanation / Answer

With n rectangles with the same width, x = (pi/2 - 0)/n = pi/2n.
f(x) = 9x cos(9x)
So, using right endpoints,
A (k = 1 to n) f(0 + kx) x
...= (k = 1 to n) f(kpi/2n ) * (pi/2n. )
...= (k = 1 to n) [9(kpi/2n ) cos(9kpi/2n) ] * (pi/2n. ).

For the exact area, let n:
A = lim(n) (k = 1 to n) [9(kpi/2n ) cos(9kpi/2n) ] * (pi/2n. ).

==>  lim(n) (k = 1 to n) [9(k^2(pi)^2/4n^2 ) cos(9kpi/2n) ]

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