Mathematics

Displacement is the total distance traveled, found by integrating velocity over time—think of it as summing up all the tiny movements $v(t) \, dt$ across the interval. Here you'll integrate $v(t) = ke^{-0.2t} + 3$ from $t=0$ to $t=10$. The key is handling the exponential term: remember that $\int e^{at} \, dt = \frac{1}{a}e^{at} + C$, so the constant $k$ and the coefficient $-0.2$ will both affect how the exponential integrates.

Displacement is the total distance traveled, found by integrating velocity over time—think of it as summing up all the tiny movements $v(t) \, dt$ across the interval. Here you'll integrate $v(t) = ke^{-0.2t} + 3$ from $t=0$ to $t=10$. The key is handling the exponential term: remember that $\int e^{at} \, dt = \frac{1}{a}e^{at} + C$, so the constant $k$ and the coefficient $-0.2$ will both affect how the exponential integrates.