Limits of integration

of a Riemann integrable function ${\displaystyle f}$ defined on a closed and bounded interval are the real numbers ${\displaystyle a}$ and ${\displaystyle b}$, in which ${\displaystyle a}$ is called the lower limit and ${\displaystyle b}$ the upper limit. The region that is bounded can be seen as the area inside ${\displaystyle a}$ and ${\displaystyle b}$.

For example, the function ${\displaystyle f(x)=x^3}$ is defined on the interval ${\displaystyle [2,4]}$ $${\displaystyle {\int }_2^4{x}^{3}dx}$$ with the limits of integration being ${\displaystyle 2}$ and ${\displaystyle 4}$.

Integration by Substitution (U-Substitution)

In Integration by substitution, the limits of integration will change due to the new function being integrated. With the function that is being derived, ${\displaystyle a}$ and ${\displaystyle b}$ are solved for ${\displaystyle f(u)}$. In general, $${\displaystyle {\int }_a^{b}f(g(x))g^{\prime }(x)dx={\int }_{g(a)}^{g(b)}f(u)du}$$ where ${\displaystyle u=g(x)}$ and ${\displaystyle du=g^{\prime }(x)dx}$. Thus, ${\displaystyle a}$ and ${\displaystyle b}$ will be solved in terms of ${\displaystyle u}$; the lower bound is ${\displaystyle g(a)}$ and the upper bound is ${\displaystyle g(b)}$.

For example, $${\displaystyle {\int }_0^{2}2x\cos (x^2)dx={\int }_0^4\cos (u)du}$$

where ${\displaystyle u=x^2}$ and ${\displaystyle du=2xdx}$. Thus, ${\displaystyle f(0)=0^2=0}$ and ${\displaystyle f(2)=2^2=4}$. Hence, the new limits of integration are ${\displaystyle 0}$ and ${\displaystyle 4}$.

The same applies for other substitutions.