Parametric Equations, Polar Coordinates, and Vector–Valued Functions (BC Only)

$\oint_C (\nabla \times \mathbf{F})\cdot d\mathbf{r}$ where $d\mathbf{r}$ lies parallel to plane $y=0$

$\oint_C (F_1,\mathrm{d}x + F_2,\mathrm{d}y + F_3,\mathrm{d}z)$

$\oint_C (\partial F_2/\partial y - \partial F_3/\partial z),\mathrm{d}z + (\partial F_3/\partial x - \partial F_1/\partial z),\mathrm{d}x$

$\int_{C}\left(\frac{\partial}{\partial x}\left(F_{1}\right)+\frac{\partial}{\partial y}\left(F_{2}\right)+\frac{\partial}{\partial z}\left(F_{3}\right)\right)dA$

Determinism guaranteed because relies on strict procedural generation of points considering every possible outcome always exact results.

Monte Carlo Integration is less effective than other techniques since randomness introduces too much variability and leads to unreliable outcomes.

Accuracy depends on random sampling density but often yields good estimates in multidimensional scenarios needing quick rough calculations.

Utilize u-substitution where u equals sin(4*t)

Implement integration by parts separating cos(4*t) differential dt

Apply partial fraction decomposition before integrating

Use symmetry properties of cosine and sine functions over given interval

Simply stating that $x$, $y$, $z$ are variables related to motion without specifying their relationship

A set of equations like $x = f_1(t)$ and $y = f_2(t)$ that define position over time

Solving differential equations unrelated to describing paths or trajectories

Using only one variable, such as $s(t)$, without breaking it down into component functions

The value of the integral $\int_1^e e^{2t} + \ln(t) + t^3 dt$.

The value of the integral $\int_1^e (2e^{2t} - \frac{1}{t^2} + 3t^2) dt$.

The value of the integral $\int_1^e \sqrt{4e^{4t} + \frac{1}{t^2} + 9t^4} dt$.

The value of the integral $\int_1^e | e^{2t} | + | \frac{1}{t}| + | t|^3 dt$.

The speed of the particle at a given time

The direction of the particle at a given time

The total change in position of the particle over a given interval

The acceleration of the particle at a given time

Integration by parts for all components.

Trigonometric substitution for the second component.

Partial fractions decomposition for all components.

U-substitution for all components.

$\langle 1 - e^0, -\sin\left(\frac{\pi}{2}\right) \rangle$

$\langle e^{-\frac{1}{2}}, \sin\left(\frac{\pi}{2}\right) \rangle$

$\langle -e^x + e^{-x}, -\cos(\pi) \rangle$

$\langle 1 - e^{-\frac{1}{2}}, 0 \rangle$

16 units

12 units

24 units

20 units

Exact integration of the derivative of each component function separately and summing them up.

Using only the first and last points to apply the Trapezoidal Rule directly on each component function.

Averaging out values obtained by Midpoint and Trapezoidal Rules applied separately on each component function.

Numerical integration techniques such as Simpson's rule applied on parametric equations after finding a formula for arc length.