a
(
t
)
=
{
200
0
≤
t
≤
5
0
e
l
s
e
a(t)=\left\{ \begin{array}{l} 200 \quad & 0 \leq t \leq 5 \\ 0 \quad & else \end{array} \right.
a(t)={20000≤t≤5else
所以，
a
(
t
)
a(t)
a(t)的无限积分为：
∫
a
(
t
)
d
t
=
A
(
t
)
=
∫
−
inf
0
0
d
t
+
∫
0
5
200
d
t
+
∫
5
+
i
n
f
0
d
t
=
1000
\int a(t) dt=A(t)= \int_{- \inf}^{0} 0 dt \; + \; \int_{0}^5 200dt + \int_{5}^{+inf} 0 dt = 1000
∫a(t)dt=A(t)=∫−inf00dt+∫05200dt+∫5+inf0dt=1000
开始推导：
∫
∫
a
(
t
)
t
d
t
d
t
=
∫
[
A
(
t
)
t
−
∫
A
(
t
)
d
t
]
d
t
=
∫
[
1000
t
−
500
t
2
]
d
t
=
500
t
2
−
500
3
t
3
\int \int a(t)t dt dt =\int \left[ A(t)t - \int A(t) dt \right] dt =\int \left[ 1000t -500t^2 \right] dt=500t^2-\frac{500}{3} t^3
∫∫a(t)tdtdt=∫[A(t)t−∫A(t)dt]dt=∫[1000t−500t2]dt=500t2−3500t3