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 $y=0$

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

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

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

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

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.

$c$

$d$

$x$

$r$

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

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

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

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

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

Divergence theorem in three dimensions.

Stokes' theorem in three dimensions.

Green's theorem in three dimensions.

Direct computation using parameterization.

Using higher-order Taylor Polynomials that do not adjust intervals but might offer more precise local approximations.

Adaptive quadrature which adjusts interval sizes dynamically based upon rate of change within intervals.

Constant-size partitioning across entire range regardless of rate of change.

Utilizing lower-order Newton-Cotes formulas that may have larger errors with rapid changes.

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

$e^{2t}$

$\ln(t)$

$t^3$

All components require integration by parts.