Just use the power rule:
[tex]\displaystyle\int_{t=-a}^{t=a}(a^2-t^2)\,\mathrm dt=a^2t-\dfrac13t^3\bigg|_{t=-a}^{t=a}[/tex]
We also could have use the fact that the integrand is even to write
[tex]\displaystyle\int_{t=-a}^{t=a}(a^2-t^2)\,\mathrm dt=2\int_{t=0}^{t=a}(a^2-t^2)\,\mathrm dt=2\left(a^2t-\dfrac13t^3\bigg|_{t=0}^{t=a}\right)[/tex]
I'll continue with this expression since it's easier to evaluate (if only a little):
[tex]2a^3-\dfrac23a^3=\dfrac43a^3[/tex]
