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\int 8t\mathrm{d}t
Evaluate the indefinite integral first.
8\int t\mathrm{d}t
Factor out the constant using \int af\left(t\right)\mathrm{d}t=a\int f\left(t\right)\mathrm{d}t.
4t^{2}
Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t\mathrm{d}t with \frac{t^{2}}{2}. Multiply 8 times \frac{t^{2}}{2}.
4\times 45^{2}-4\times 0^{2}
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.
8100
Simplify.