Evaluating an integral from a Fourier series when $f$ is defined piecewise

Question

Given that: $$f(t)= \begin{cases} -1\space if & t<0 \\2\space if & t>=0 \end{cases}$$

I have to evaluate the following integral \begin{equation} a0=\frac{1}{T}\int_{-T/2}^{T/2}f(t)dt \end{equation} How do I start solving this? Thanks

Answer 1

How do I start solving this?

Split the integral into $$ \int_{-T/2}^0 f(t) dt + \int_0^{T/2} f(t) dt.$$ In each integral, you know what $f(t)$ is explicitly. Substitute this value in and evaluate the integrals.

Answer 2

This is just basic calculus

$$\int_{-T/2}^0 -1 + \int_0^{T/2}2 = {T\over 2} + 2{T\over 2} = {3\over 2}T$$

now divide by $T$ to get $3/2$.