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

$\int_{0}^{\pi} t \sqrt{e^{-2t}} dt$

$\int_{0}^{\pi} \sqrt{e^{-2t}(1-t^2)} dt$

$\int_{0}^{\pi} e^{-2t} dt$

$\int_{0}^{\pi} \sqrt{e^{-2t}} dt$

Both $\frac{dx}{dt}$ and $\frac{dy}{dt}$

Only $\frac{dy}{dt}$

Neither derivative is necessary

Only $\frac{dx}{dt}$

$\sqrt{2}$

$4\sqrt{2}$

$3\sqrt{2}$

$2\sqrt{2}$

Ratio Test

Integral Test

Root Test

Alternating Series Test

Comparison Test with $\frac{1}{k^{3/2}}$ or another suitable p-series or function where p > 1

Geometric Series Test due to its form similar to geometric progression with variable ratios between terms

Limit Comparison Test using a comparable harmonic series would indicate divergence because both have similar forms but different rates of growth or decay in their terms' sizes related to n-values at large k limits within domain constraints imposed on this specific context under evaluation here today right now indeed surely yes absolutely certainly without doubt no question about it hooray hurrah whoopee yippee yay woohoo!

Divergence Test because each term does not approach zero as $k$ approaches infinity

2 + 2\sqrt{2}

1 + \sqrt{2}

2 + \sqrt{2}

3 + 2\sqrt{2}

The maximum value reached by parameter $t$

The starting value for parameter $t$

The rate at which parameter $t$ changes

The ending value for parameter $t$

$\int \left( \frac{dx}{dt} - \frac{dy}{dt} \right) dt$

$\int \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } dt$

$\int \left( \frac{dx}{dt} + \frac{dy}{dt} \right) dt$

$\int \left( \frac{dx}{dt} \times \frac{dy}{dt} \right) dt$

The arc length will decrease due to the compression on the x-axis.

The arc length will increase but not quadruple.

The arc length will remain unchanged.

The arc length will be multiplied by 4.

Both functions must be continuous on [a,b]