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!
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!
Answers
MathMate
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
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