Q: Which of the following is the correct formula for the area $$A$$ enclosed by a polar curve $$r = f(\theta)$$ from $$\theta = a to$$ $$\theta = b$$?

$$A = \frac{1}{2} \int_a^b [f(\theta)]^2 d\theta$$

Q: Find the area enclosed by one loop of the rose curve $$r = 2 \cos 3\theta.$$

$$A = \$$int_0$$^{\frac{\pi}{3}} \frac{1}{2} [2 \cos 3\theta]^2 d\theta$$

Q: If a region $$R in $$polar coordinates is bounded by $$r = 2$$ and $$r = 4$$, for $$0 \leq \theta \leq \frac{\pi}{2}$$, what is the area of $$R$$?

$$A = \frac{1}{2} \$$int_0$$^{\frac{\pi}{2}} (16 - 4) d\theta$$

Q: Which of the following is the correct conversion from rectangular to polar coordinates?

$$x = r \cos \theta, \quad y = r \sin \theta$$

Question 1

Given the parametric equations $$x = 4 \cos t$$ and $$y = 4 \sin t$$, for $$0 \leq t \leq 2\pi$$, what is the geometric shape described by these equations?

Accepted Answer

A circle of radius 4 centered at the origin

Question 2

For the parametric curve $$x = 4 \cos t$$, $$y = 4 \sin t$$, at which values of $$t$$ does the curve have a vertical tangent?

Accepted Answer

$$t = 0, \pi$$

Question 3

What is the formula for the arc length $$L of a $$parametric curve $$x = f(t)$$, $$y = g(t)$$ for $$a \leq t \leq b$$?

Accepted Answer

$$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\$$right)^2$$ + \left(\frac{dy}{dt}\$$right)^2$$} \, dt$$

Question 4

Convert the polar equation $$r = 4 \sin 	heta to $$rectangular coordinates.

Accepted Answer

$$x^2$$ + $$(y-2)^2 = 4$

Question 5

Which of the following is the correct formula for the area $$A$$ enclosed by a polar curve $$r = f(	heta)$$ from $$	heta = a to$$ $$	heta = b$$?

Accepted Answer

$$A = \frac{1}{2} \int_a^b [f(	heta)]^2 d	heta$$

Question 6

Find the area enclosed by one loop of the rose curve $$r = 2 \cos 3	heta.$$

Accepted Answer

$$A = \$$int_0$$^{\frac{\pi}{3}} \frac{1}{2} [2 \cos 3	heta]^2 d	heta$$

Question 7

Given the polar curve $$r = 2 + 3 \sin 	heta$$, what is the slope $$\frac{dy}{dx} at$$ $$	heta = 0$$?

Accepted Answer

$$0$$

Question 8

Which of the following is the standard equation of a parabola opening upward with vertex at the origin?

Accepted Answer

$$x^2 = 4py$

Question 9

For the ellipse $$\frac{x^2$}{9} + \frac{y^2$}{4} = 1$$, what is the length of the major axis?

Accepted Answer

$$6$$

Question 10

Which of the following best describes the geometric definition of a hyperbola?

Accepted Answer

The set of all points where the absolute value of the difference of the distances to two fixed points (foci) is constant.

Question 11

Given the ellipse $$\frac{x^2$}{16} + \frac{y^2$}{4} = 1$$, what is the distance between the foci?

Accepted Answer

$$4\sqrt{3}$$

Question 12

If a region $$R in $$polar coordinates is bounded by $$r = 2$$ and $$r = 4$$, for $$0 \leq 	heta \leq \frac{\pi}{2}$$, what is the area of $$R$$?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\frac{\pi}{2}} (16 - 4) d	heta$$

Question 13

Which of the following is the correct conversion from rectangular to polar coordinates?

Accepted Answer

$$x = r \cos 	heta, \quad y = r \sin 	heta$$

Question 14

For the hyperbola $$\frac{x^2$}{9} - \frac{y^2$}{4} = 1$$, what are the equations of the asymptotes?

Accepted Answer

$$y = \pm \frac{2}{3}x$$

Question 15

A circle in polar coordinates is given by $$r = 3. $$What is its equation in rectangular coordinates?

Accepted Answer

$$x^2$$ + $$y^2 = 9$

Question 16

Given the parametric equations $$x = t^2$, y$$ = $$t^3$$, find $$\frac{dy}{dx} in $$terms of $$t.$$

Accepted Answer

$$\frac{dy}{dx} = \frac{3t^2$}{2t}$$

Parametric and Polar Curves, Arc Length, Area, and Conic Sections

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