Suppose $f$ is a continuous function.

$($a$)$ Find $f(4)$ given ${\displaystyle \underset{0}{\overset{x^2}{}}}f(t)dt=xsin(x)$ for all $x.$

$($b$)$ Find $f(4)$ given ${\displaystyle \underset{0}{\overset{f(x)}{}}}{t}^{2}dt=xsin(x)$ for all $x.$