What's wrong with this change of variable in calculating definite integral?

Question

Suppose we want to calculate $S=\int_{0}^{1}(-x^2+x)dx$ and somehow make the change of variable $x=-t$, so $S=\int_{-1}^{0}(-t^2-t)(-dt)=\int_{-1}^{0}(t^2+t)dt=(\frac{t^3}{3}+\frac{t^2}{2})\big|_{-1}^{0}=0-(-\frac{1}{3}+\frac{1}{2})=-\frac{1}{6}$. This is obviously incorrect because the original integral $\int_{0}^{1}(-x^2+x)dx=(-\frac{x^3}{3}+\frac{x^2}{2})\big|_{0}^{1}=\frac{1}{6}$. Which step is wrong in this change of variable procedure, and why? Thanks!

Answer 1

The limits are wrong.

When $~x~$ is $~0~$, $~t~$ is still $~0~$.

When $~x~$ is $~1~$, $~t=-1~$

So integral is from $~0~$ to $~-1~$ and not form $~-1~$ to $~0~$.