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 example, 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