Integration:
Compute the following integrals:
- $$\int \langle \sin t, 2t, -8t^3 \rangle dt $$
- $$\int \langle e^{\cos t} \sin t, t^2 \sin t, t^2 \tan t^3 \rangle dt$$
- $$\int_1^4 \langle \sqrt{t}, t e^{-t}, t^{-2} \rangle dt $$
More Integration:
Find $\mathbf{r}(t)$ if $\mathbf{r}'(t) = \langle t, e^t, t e^t \rangle$ and $\mathbf{r}(0) = \langle 1, 1, 1 \rangle.$:
Velocity and Acceleration:
An object is moving along the curve $\mathbf{r}(t) = \langle t, t^3, 3t \rangle$ for $t \ge 0$.
Find the velocity, $\mathbf{v}(t)$, and acceleration, $\mathbf{a}(t)$, when $t = 1$.
Velocity and Acceleration:
A moving particle starts at position $\mathbf{r}(0) = \langle 1, 0, 0 \rangle$ with initial velocity $\mathbf{v}(0) = \langle 1, -1, 1 \rangle$. Its acceleration is $\mathbf{a}(t) = \langle 4t, 6t, 1 \rangle$. Find its velocity and position functions at time $t$.