Question

General question about definite integrals

In my calculus textbook, they seem to imply (although never explicitly say) that the lower bound should be lower in value than the upper bound when evaluating definite integrals. So for example, if asked to solve \[\int_2^0 x^2\ dx\] they would rearrange the integral like this and then solve:

\[\int_2^0 x^2\ dx = -\int_0^2 x^2\ dx\]
\[= \left. -\frac{x^3}{3}\right]_0^2\]
\[F(b) - F(a) = \frac{2^3}{3} - \frac{0^3}{3} = -\frac{8}{3}\]

Answer 1

If I solve this without rearranging the integral, though, I get the same exact answer. Why is it necessary to rearrange the upper and lower bounds?

Answer 2

it isn't

Answer 3

well thank you sir