# Formula of magnetic flux through a loop as a function of time

I was studying Lenz's law and self inductance when I came across the following question.

Take R to be the radius of a circular loop, which has a magnetic field perpendicular to this loop. We let the magnitude of the magnetic field be $$B=at^2r^2$$ Derive a formula for the magnetic flux through the loop as a function of time.

I intially used the formula for it, (the integral of the dot product of B.dA) and I get $$\pi a R^4 t^2$$ but the answer is meant to be $$\frac{\pi aR^4t^2}{2}$$

Im not sure where the $$\frac{1}{2}$$ comes from. Any help would be appreciated, thank you.

$$\psi=\int at^2r^2dA=\int^{2\pi}_0\int^R_0at^2r^2rdrd\theta$$
$$at^2\int^{2\pi}_0d\theta\int^R_0r^3dr=at^2*\theta|^{2\pi}_0*\frac{r^4}{4}|^R_0=at^2*2\pi*\frac{r^4}{4}$$
which equals $$\frac{\pi at^2r^4}{2}$$