Area of a Polar Region - AP Calculus BC

1. What is the differential area element in polar coordinates?

   $dA = \frac{1}{2}r^2 d\theta$. Infinitesimal area element in polar form.

2. Identify the polar equation for a circle centered at the origin with radius $a$.

   $r = a$. Constant radius from origin defines a circle.

3. Which polar equation represents a dimpled limaçon?

   $r = a + b\text{cos}\theta$, $a > b$. When $a > b$, no inner loop forms.

4. State the formula for converting angular coordinates to polar.

   $\theta = \text{tan}^{-1}(\frac{y}{x})$. Angular coordinate conversion from Cartesian.

5. Convert the polar point $(r, \theta)$ to Cartesian coordinates.

   $(x, y) = (r\text{cos}\theta, r\text{sin}\theta)$. Standard polar to Cartesian conversion.

6. What is the value of $\theta$ for a polar axis?

   $\theta = 0$. The positive x-axis corresponds to $\theta = 0$.

7. Which polar equation represents a limaçon without an inner loop?

   $r = a + b\text{cos}\theta$, $a = b$. Boundary case between loop and no loop.

8. State the condition for a polar curve having symmetry about the origin.

   Replace $\theta$ with $\theta + \text{pi}$; identical equation. Alternative test for origin symmetry.

9. Identify the polar equation for an ellipse.

   $r = \frac{a}{1 + e \cos \theta}$, $e < 1$. Conic with eccentricity less than 1.

10. Which polar function represents a cardioid?

    $r = a(1 + \text{cos}\theta)$. Heart-shaped curve with one cusp.

11. What defines the symmetry about the origin for a polar curve?

    Replace $r$ with $-r$; identical equation. Point reflection through the origin.

12. What is the polar equation for a lemniscate?

    $r^2 = a^2 \text{cos}(2\theta)$. Figure-eight shaped curve.

13. What is the polar equation for a parabola?

    $r = \frac{a}{1 + e \cos \theta}$, $e = 1$. Conic with eccentricity exactly 1.

14. Which polar equation represents a limaçon with an inner loop?

    $r = a + b\text{cos}\theta$, $a < b$. When $a < b$, creates inner loop.

15. State the general form of a polar equation.

    $r = f(\theta)$. Most general form of polar equations.

16. State the symmetry about the polar axis condition for a polar curve.

    Replace $\theta$ with $-\theta$; identical equation. If curve unchanged when $\theta$ becomes $-\theta$.

17. Identify the range of $r$ for a polar region bounded by $r = 1 + \text{cos}\theta$.

    $0 \text{ to } 2$. Cardioid ranges from minimum 0 to maximum 2.

18. State the symmetry about the line $\theta = \frac{\text{pi}}{2}$ for a polar curve.

    Replace $\theta$ with $\text{pi} - \theta$; identical equation. Reflection across the line $\theta = \frac{\pi}{2}$.

19. What is the polar equation for a line through the pole?

    $\theta = \text{constant}$. Ray from origin at fixed angle.

20. What is the polar equation for a spiral of Archimedes?

    $r = a\theta$. Radius increases linearly with angle.

21. Which polar equation represents a rose curve?

    $r = a\text{cos}(n\theta)$ or $r = a\text{sin}(n\theta)$. Creates petals, number depends on $n$.

22. Convert Cartesian coordinates $(x, y)$ to polar coordinates.

    $(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Standard Cartesian to polar conversion.

23. What is the polar equation for a hyperbola?

    $r = \frac{a}{1 + e \cos \theta}$, $e > 1$. Conic with eccentricity greater than 1.

24. State the form of a polar equation for a conic section.

    $r = \frac{ed}{1 + e\text{cos}\theta}$. General conic section in polar coordinates.

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