Applications of Integration

Consider a particle moving along a path. The position function of the particle is given by $s(t) = 2t^3 - 3t^2 + 4t + 1$. What is the acceleration function of the particle?

Given a position function defined by $s(t) = t^3 - \frac{9}{2} t^2 +4$, what is the total distance traveled by an object from time $t=1$ to time $t=3$?

If the position function $s(t)$ of an object is given by the integral of its velocity function $v(t)=t^2-4t+3$, what is the best method to find the object's position at time $t=5$ if it starts from rest at the origin?

Consider a particle moving along a path. The velocity function of the particle is $v(t) = 3e^t - 2t$. What is the position function of the particle?

If the velocity function $v(t) = 3t^2 - 6t$ represents the velocity of an object moving along a line, what is the object's acceleration at time $t = 2$?

If an object's position as a function of time is given by $x(t)=4e^{-0.5t}$ for times greater than zero and its initial velocity was zero when at rest ($x=0$), how would you represent its acceleration at any time point ($A(t)$)?

If the velocity of an object is given by the function $v(t) = 3t^2 - 4t$, what is the position function $s(t)$ if the initial position is $s(0) = 5$?

What does it mean when a position function, represented as $s(t)$, has a constant second derivative?

Considering the parametric equations $(x(t), y(t))=(te^{-kt}, t^{n}), k>0, n \in \mathbb{N}$, how should n vary relative to k in order for the curve described by x over y as a function of time $(x/y)(t)$ to demonstrate concave up behavior on interval $(m,o), (m>o)?$

Consider a particle moving along a path. The velocity function of the particle is given by $v(t) = 2t - 5$. What is the acceleration function of the particle?