Question

I am trying to prepare for my finals and I'm approached with this question where I'm not entirely clear what the answer should be:

Given the function h(t) graphed below, define A(x) = ∫[-0.5,x] h(t) dt .

(Graph is in the link)
goo.gl/photos/YZuHtnUK2UPwpaq1A

Evaluate A(0.5):

=∫[-0.5,0.5] h(t) dt

=h(0.5) - h(-0.5)

= -1 - 0

= -1 is the answer?

Any help is greatly appreciated!

Answer 1

Function h is a piece-wise continuous function, i.e. the function is defined the combination a number of segment, each of which is a different mathematical function.

For example, we can consider
h(t)={0 for t<0, 1 for 0≤t<0.5, -1 for 0.5≤t<1, 0 for t≥1}

You can therefore integrate the function segment by segment (if the limits cross segment boundaries), and add together the integral of the segments.

For <i>example</i>, to evaluate the integral between limits (-∞, ∞), we would find
I1=0
I2=0.5
I3=-0.5
I4=0
so the integral=I1+I2+I3+I4=0