Determine the area of the given region under the curve. y=1/x^3 http://www.webassign.net/larson/4_04-36.gif

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You need to determine the area of the region under the curve of the given function `f(x) = y = 1/x^3` , over the interval `[1,2] ` such that:

`A = int_1^2 f(x) dx`

Substituting `1/x^3`  for `f(x)`  yields:

`A = int_1^2 1/x^3 dx = int_1^2 x^(-3) dx`

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You need to determine the area of the region under the curve of the given function `f(x) = y = 1/x^3` , over the interval `[1,2] ` such that:

`A = int_1^2 f(x) dx`

Substituting `1/x^3`  for `f(x)`  yields:

`A = int_1^2 1/x^3 dx = int_1^2 x^(-3) dx`

You need to use the fundamental theorem of calculus to evaluate the definite integral such that:

`int_(x_1)^(x_2) f(x) dx = F(x_2) - F(x_1)`

`A = x^(-3+1)/(-3+1)|_(1)^(2) => A = -1/(2x^2)|_1^2`

`A = -1/(2*2^2) + 1/(2*1^2)`

`A = -1/8 + 1/2 => A = (-1+4)/8 => A = 3/8`

Hence, evaluating the area under the given curve, over the interval `[1,2], ` yields `A = 3/8` .

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