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

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.

$\sqrt{2}$

$4\sqrt{2}$

$3\sqrt{2}$

$2\sqrt{2}$

Ratio Test

Integral Test

Root Test

Alternating Series Test

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 $t$

Circle undergoes numerous cycles compared to two single spiral turns pre-intersection caused by greater perimeter relation versus expanding radius effectivity delaying convergence until later rounds.

They meet after both complete exact one revolution since spiral's growth rate guarantees intersection within single consistent cycle per definition.

They don't overlap until after second revolution because radial expansion outpaces angular progression thereby requiring additional rotation before intersecting.

Spiral completes more rotations due to relative proportional growth rates whereas circular motion remains constant throughout allowing several encounters beforehand.

2 + 2\sqrt{2}

1 + \sqrt{2}

2 + \sqrt{2}

3 + 2\sqrt{2}

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

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

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

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

Both functions must be continuous on [a,b]

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

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

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

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

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

Only $\frac{dy}{dt}$

Neither derivative is necessary

Only $\frac{dx}{dt}$