$$\int x^{3}\mathrm{d}x$$
Since $\int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1}$ for $k\neq -1$, replace $\int x^{3}\mathrm{d}x$ with $\frac{x^{4}}{4}$.$$\frac{x^{4}}{4}$$
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.$$\frac{b^{4}}{4}-\frac{0^{4}}{4}$$
Simplify.$$\frac{b^{4}}{4}$$