How to integrate a piecewise function?

Questions
,
1

I keep getting funny errors when trying to do this:

g[t_] := -9.8;
a[t_] := Piecewise[{{0, t < 0}, {12, 0 <= t < 1}, {14, 
     1 <= t < 3}, {11, 3 <= t < 8}, {0, t >= 8}}];
Plot[a[t], {t, 0, 10}]
v[t_] := Integrate[a[t], t];
v[t]
Integrate[a[t], t]
Plot[v[t], {t, 0, 10}]
Plot[g[t], {t, 0, 10}]

Piecewise Results.JPG
Piecewise Results.JPG928×721 38.7 KB

2

When you definve v[t_]:=Integrate [f[t],t] and you try to plot it you are basically solving for each t

Plot[{
    Integrate[f[0],0] 
    Integrate[f[1],1] 
    Integrate[f[2],2]
    ...
}]

That won’t work, so first calculate the integral and then define a function that replaces after integration.

vt = Integrate[a[t], t] 
v[ti_] := vt /. t -> ti
Plot[v[t], {t, 0, 10}]
3

Thank you very much!! I don’t know how I could have divined this from the documentation. What is the last line (117 Ture) trying to tell me?
I would now like to integrate v[t]. Do I proceed in the same way?`

g[t_] := -9.8;
a[t_] := Piecewise[{{0, t < 0}, {12, 0 <= t < 1}, {25, 
     1 <= t < 3}, {11, 3 <= t < 8}, {0, t >= 8}}];
Plot[a[t], {t, 0, 10}]
v1 = Integrate[a[t], t];
v2[tj_] := v1 /. t -> tj
v2[t]
Plot[v2[t], {t, -1, 10}]

vt = Integrate[a[t], t];
v[ti_] := vt /. t -> ti;
v[t]
Plot [v[t], {t, 0, 10}]
Plot[g[t], {t, 0, 10}]

Piecewise Results 2.JPG
Piecewise Results 2.JPG753×859 28.3 KB

4

Hi Orest,

An alternative to Martijn’s answer

(* Acceleration *)
a[t_] := 
 Piecewise[{{0, t < 0}, {12, 0 <= t < 1}, {25, 1 <= t < 3}, {11, 3 <= t < 8}, {0, t >= 8}}]

(* Velocity *)
v[t_] := Integrate[a[t], t];

(* Distance *)
s[t_] := Integrate[v[t], t];

Plot[Evaluate[{a[t], v[t], s[t]}], {t, 0, 10},
 PlotTheme -> "Detailed",
 PlotRange -> All,
 ImageSize -> 600]

10964plot.png
10964plot.png812×381 28.6 KB

The 117 True in the Piecewise means the value is 117 everywhere else in the domain t > 8.

5

I tried this first and was getting errors. Obviously, you don’t. Wonder why.
I’ll try again.
Thank you!

GOGO