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Use the Definition to find an expression for the area under the graph of f as a limit....

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Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit

f(x) = x^2+ sqrt(1+2x)   6 ≤ x ≤ 8

 

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You should find the area under the given curve, hence, you need to evaluate the limit of sum of areas of small rectangles as the number of rectangles under the curve approaches to infinite.

You need to consider the width of rectangle as `Delta x` and the height of rectangle as `f(x_i)` .

You need to evaluate in case of n rectangles such that:

`Delta x = (8-6)/n => Delta x = 2/n`

You need to evaluate the area of rectangle using the right points such that:

`A = f(x_i)*Delta x`

`x_i = 6 + iDelta x => x_i = 6 + 2i/n`

Substituting `x_i` in `f(x_i)` yields:

`f(x_i) = (6 + 2i/n)^2 + sqrt(1 +12 + 4i/n)`

`A = f(x_i)*Delta x = ((6+2i/n)^2 + sqrt(13+4i/n))*(2/n)`

Hence, evaluating the area under the curve as the number of rectangles approaches to infinite yields:

`A = lim_(n->oo) sum_(i=1)^n (72/n + 24i/n^2 + 8i^2/n^3 + (2/n)sqrt(13+4i/n))` 

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