Question

Help plz! Evaluate the following definite integral:
`\int\limits^a_(-a) {(a^(2) - t^(2) )} \, dt`

Answer 1

Just use the power rule:

`\displaystyle\int_(t=-a)^(t=a)(a^2-t^2)\,\mathrm dt=a^2t-\frac13t^3\bigg|_(t=-a)^(t=a)`

We also could have use the fact that the integrand is even to write

`\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-\frac13t^3\bigg|_(t=0)^(t=a)\right)`

I'll continue with this expression since it's easier to evaluate (if only a little):

`2a^3-\frac23a^3=\frac43a^3`