Answer:
0.44237 units
Step-by-step explanation:
Given
[tex]v(t) = e^t*sin(e^t)[/tex]
[tex]0 \le t \le \frac{\pi}{2}[/tex]
Required
The distance traveled in this interval
We have:
[tex]v(t) = e^t*sin(e^t)[/tex]
The distance is calculated as:
[tex]d(t) = \int\limits^a_b {v(t)} \, dt[/tex]
So, we have:
[tex]d(t) = \int\limits^{\frac{\pi}{2}}_0 {e^t*sin(e^t)} \, dt[/tex]
Let:
[tex]u = e^t[/tex]
Differentiate
[tex]\frac{du}{dt} = e^t[/tex]
So:
[tex]dt = e^{-t} du[/tex]
So, we have:
[tex]d(t) = \int\limits^{\frac{\pi}{2}}_0 {e^t*sin(e^t)} \, dt[/tex]
[tex]d(t) = \int\limits^{\frac{\pi}{2}}_0 {e^t*sin(u)} \, e^{-t} du[/tex]
Rewrite as:
[tex]d(t) = \int\limits^{\frac{\pi}{2}}_0 {e^t* e^{-t}*sin(u)} \, du[/tex]
[tex]d(t) = \int\limits^{\frac{\pi}{2}}_0 {sin(u)} \, du[/tex]
Integrate:
[tex]d(t) = -\cos(u)|\limits^{\frac{\pi}{2}}_0[/tex]
Substitute [tex]u = e^t[/tex]
[tex]d(t) = -\cos(e^t)|\limits^{\frac{\pi}{2}}_0[/tex]
Split
[tex]d(t) = -\cos(e^\frac{\pi}{2}) - [-\cos(e^0)][/tex]
[tex]d(t) = -\cos(e^\frac{\pi}{2}) +\cos(e^0)[/tex]
[tex]d(t) = -0.09793 + 0.5403[/tex]
[tex]d(t) = 0.44237[/tex]
The distance traveled in this interval is: 0.44237 units
dt =
d(t) = \int\limits^2_0 {e^t*sin(e^t)} \, dt
