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) at 0 ≤ x ≤ pi/2

Expert Answers

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You need to split the area under the graph in n rectangles that have the widths `Delta x`  and the heights`f(x_i).`

You need to evaluate the area of each rectangle such that:

`A = f(x_i)*Delta x`

Adding all these areas of the rectangles yields:

Area = `sum_(i=1)^n``f(x_i)*Delta x`

You need to evaluate Delta x over interval `[0,pi/2]`  such that:

`Delta x = (pi/2 - 0)/n = pi/(2n)`

Using the right endpoints, you may evaluate `f(x_i)`  such that:

`x_i= 0 + i*Delta x = i*pi/(2n)`

`f(x_i) = 9*x_i*cos(9x_i)`

`f(x_i)*Delta x = 9i*pi/(2n)*pi/(2n)*cos(9i*pi/(2n))`

Hence, evaluating the area using limit definition yields:

Area = `lim_(n-gtoo)`  `sum_(i=1)^n` `9i(pi^2)/(4n^2)cos(9i*pi/(2n))`

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