discuss the fundamental theorem of classical analysis.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember the fundamental theorem of calculus, hence, you need to consider the function `f(x)` , that is continuous over the closed interval `[a,b]` and `F(x)` , that is the primitive of the function `f(x)` , such that:

`int_a^b f(x)dx = F(x)|_a^b`

`F(x)|_a^b = F(b) - F(a)`

The fundamental theorem of calculus has a geometric meaning, hence, this theorem allows you to evaluate the area of a region between the curve `f(x)` , the x axis and the lines `x = a, x = b` .

Consider the following example that requests for you to evaluate the area of the region enclosed by the curve `f(x)=x` , the x axis, over the interval `[1,2]` , such that:

`A = int_1^2 x dx => A = x^2/2|_1^2`

`A = 2^2/2 - 1^2/2 => A = 4/2 - 1/2 => A = 2 - 1/2 => A = 3/2`

Hence, the fundamental theorem of calculus helps you to evaluate the definite integrals and the area of a region enclosed by some given curves, over an interval `[a,b].`