Let f(x) be a continuous function and consider the integration of f(x) from x =

Question

Let f(x) be a continuous function and consider the integration of f(x) from x = a to x = b where b > a. It is easily shown using the basic definition of the Riemann integral (the one we are familiar with) that I_a rightarrow b = integral ^b _a f(x)dx = - I_b rightarrow a. where I_b rightarrow a = integral ^a_b f(x)dx.Also let X_n, n = 1, 2,...N by N points lying between a and b with x_1 = a x_n = b and consider the sum S_a rightarrow b = sigma^N _ n = 1f(x_n). S_a rightarrow b = S_b rightarrow a = sigma^1 _n = N f(x_n). Thus, the sign of the integral of a function changes when the direction of integration changes but the sign of a sum does not change when we reverse the order of summation. The question is why is this so since integration is basically a summing operation

Explanation / Answer

Sum is actually the area between the curve and the x axis between a and b.

As area is always positive, the sum whether b to a or a to b is always positive.

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