What happens when we reverse the limits of a definite integral?

When you reverse the limits of a definite integral, the sign of the integral changes. Mathematically, if you have a definite integral of a function $f(x)$ from $a$ to $b$:

$\int_{a}^{b} f(x) \,dx$

and you reverse the limits to $b$ and $a$:

$\int_{b}^{a} f(x) \,dx$

the new integral is equal to the negative of the original integral:

$\int_{b}^{a} f(x) \,dx = -\int_{a}^{b} f(x) \,dx$

This is a consequence of the fact that the definite integral represents the signed area under the curve of the function between the given limits. Reversing the limits effectively changes the orientation of the interval, resulting in a change of sign for the area.

In summary, reversing the limits of a definite integral changes the sign of the integral.