650 Wars. 
pt! hr at 
A(t!) “ near | B(ey VSN aR aKtateS 
ry ° Jo ° 
Interchanging the order of integration and integrating over t, we ob- 
tain 
b! ie 
A(tt)2 | F(x) | S'(t = x) dt dx 
fo) Jo 
i 
(8) ‘) 
= | S(t! =x) F(x)dx, if S(0) <0. 
ie) 
‘ 
U 
If the applied function F(x) becomes negligible in a time somewhat less 
than tit and the step response is essentially flat over the remaining inter— 
val, the area A(t') is determined essentially by the limiting value of the 
step response at long time. For example, if the step response S(t) has a 
limiting, or shoulder, value S(oo) as t +a, and the 2pplied function is 
significant over a limited time range, the appropriate response value for 
use in area calculations is S(m). The limiting response area under the 
response curve as ah approaches infinity is 
fe 
(9) A(co) = Lim S(tt — x) F(x) dx. 
t!— 00 
Te) 
If interchanging the integration and limiting processes is legitimate, 
[8 6) 
(10) A(@) = | Lim S(t! - x) F(x) dx. 
Jo t!>@ 
For functions S(X) and F(X), which converge sufficiently rapidly to limiting 
values S(@) and zero, respectively, as X approaches infinity, this becomes 
wo 
A({w) = S(a | F(x) dx. 
Jo 
