change of variables in integral - how do limits change?

Question

If I have this integral:

$$\int_0^{\sigma_0}xR^2dR$$

and I know that $x=\frac{R} {\sigma_0}$

and I substitute:

$$\int x (x\sigma_0)^2 dx= \int x^3 \sigma_0^2 dx$$

what are the new integration limits?

Answer 1

The limits of integration are given as $R= 0$ and $R= \sigma_0$. You change variables from R to $x= \frac{R}{\sigma_0}$. When $R= 0$, then, $x= \frac{0}{\sigma_0}= 0$. When $R= \sigma_0$, $x= \frac{\sigma_0}{\sigma_0}= 1$

Answer 2

Simply replace the integration limits according to the substitution. The new extremes turn out to be $0$ and $1$.