Area Bounded By Two Polar Curves - AP Calculus BC

1. For $r = 4\cos(\theta)$ and $r = 4\sin(\theta)$, find the intersection points.

   $\theta = \frac{\pi}{4}, \frac{5\pi}{4}$. Found by solving $4\cos(\theta) = 4\sin(\theta)$, so $\tan(\theta) = 1$.

2. Which polar curve represents a cardioid: $r = 1 + \cos(\theta)$ or $r = 2\sin(\theta)$?

   $r = 1 + \cos(\theta)$. Heart-shaped curve with cusp at origin.

3. What does the expression $r = a(1 \pm \cos(\theta))$ represent?

   A cardioid. Heart-shaped curve created when $a = b$ in limaçon equation.

4. For $r = 2 + \cos(\theta)$, what is the maximum value of $r$?

   $r = 3$. Maximum occurs when $\cos(\theta) = 1$, so $r = 2 + 1 = 3$.

5. How do you convert a polar area integral to Cartesian form?

   Use $x = r\cos(\theta)$, $y = r\sin(\theta)$. Uses standard polar-to-Cartesian coordinate transformation.

6. What is the polar area formula if $g(\theta) = 0$?

   $A = \frac{1}{2} \int_{\alpha}^{\beta} f(\theta)^2 \, d\theta$. Reduces to the single curve area formula when inner curve is zero.

7. State the formula for the area between two polar curves $r = f(\theta)$ and $r = g(\theta)$.

   $A = \frac{1}{2} \int_{\alpha}^{\beta} (f(\theta)^2 - g(\theta)^2) \, d\theta$. Subtracts the inner curve's area from the outer curve's area.

8. What is the first step in finding the area between two polar curves?

   Determine points of intersection. Essential to establish integration limits for the area calculation.

9. Determine the area between $r = 1 + \sin(\theta)$ and $r = 1$ for $\theta = 0$ to $\theta = \pi$.

   $\frac{1}{2} \int_{0}^{\pi} ((1 + \sin(\theta))^2 - 1) \, d\theta$. Cardioid minus circle area using difference formula.

10. What is the integral expression for the area inside $r = 3\sin(\theta)$ but outside $r = 1$?

    $\frac{1}{2} \int_{\alpha}^{\beta} (9\sin^2(\theta) - 1) \, d\theta$. Region between circle and line using difference formula.

11. State the general steps to calculate the area between two polar curves.

    Find intersections, set up integral, evaluate. Standard procedure: intersections, integral setup, evaluation.

12. What is the area enclosed by $r = 2 + 3\cos(\theta)$ from $\theta = 0$ to $\theta = \pi$?

    $\frac{1}{2} \int_{0}^{\pi} (2 + 3\cos(\theta))^2 \, d\theta$. Limaçon area formula integrated over half period.

13. Which function describes a limaçon: $r = a \pm b\sin(\theta)$ or $r = a\cos(\theta)$?

    $r = a \pm b\sin(\theta)$. Limaçon equation includes both sine and cosine variations.

14. What type of polar curve is defined by $r = a \sin(n\theta)$?

    Rose curve. Petal curves with $n$ petals when $n$ is odd.

15. Determine the area inside $r = 1$ but outside $r = 0.5$.

    $\frac{1}{2} \int_{0}^{2\pi} (1 - 0.25) \, d\theta$. Annular region between two concentric circles.

16. Identify the area enclosed by $r = 1 - \cos(\theta)$ from $\theta = 0$ to $\theta = \pi$.

    $A = \frac{1}{2} \int_{0}^{\pi} (1 - \cos(\theta))^2 \, d\theta$. Cardioid area formula integrated over half period.

17. What is the result of $\int_{0}^{\pi} \sin^2(\theta) \, d\theta$?

    $\frac{\pi}{2}$. Uses the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$.

18. Calculate the complete area enclosed by $r = 2\cos(\theta)$.

    $A = \frac{1}{2} \int_{0}^{2\pi} (2\cos(\theta))^2 \, d\theta$. Circle area formula integrated over full period.

19. What is the area of one petal of the rose curve $r = 2\sin(2\theta)$?

    $\frac{1}{2} \int_{0}^{\frac{\pi}{2}} (2\sin(2\theta))^2 \, d\theta$. Four-petal rose with area of one petal calculated.

20. Determine the area between $r = 3\cos(\theta)$ and $r = 1$ from $\theta = 0$ to $\theta = \pi$.

    $\frac{1}{2} \int_{0}^{\pi} (9\cos^2(\theta) - 1) \, d\theta$. Circle minus circle area using difference formula.

21. What does $d\theta$ represent in the polar area integral?

    A small change in the angle $\theta$. Represents an infinitesimal angular increment in polar coordinates.

22. Determine which curve is outer: $r = 3 + 2\sin(\theta)$ or $r = 2$?

    $r = 3 + 2\sin(\theta)$ is outer. Limaçon has larger radius values than the constant circle.

23. Which polar curve is a circle: $r = 2$ or $r = 3\sin(\theta)$?

    $r = 2$ is a circle. Constant radius creates a circle centered at origin.

24. Identify the area enclosed by $r = 2 - 2\sin(\theta)$ from $\theta = 0$ to $\theta = 2\pi$.

    $A = \frac{1}{2} \int_{0}^{2\pi} (2 - 2\sin(\theta))^2 \, d\theta$. Cardioid area formula integrated over full period.

25. State the formula to convert $r = f(\theta)$ to Cartesian coordinates.

    $x = r\cos(\theta)$, $y = r\sin(\theta)$. Standard polar-to-Cartesian transformation formulas.

26. Identify the region type for $r = a \pm b\cos(\theta)$ when $a < b$.

    Limaçon with an inner loop. When inner coefficient exceeds outer, creates inner loop.

27. What is the polar area formula for the circle $r = a \cos(\theta)$?

    $A = \frac{1}{2} \int_{0}^{\pi} (a\cos(\theta))^2 \, d\theta$. Circle formula integrated over half period.

28. Find the area inside $r = 4\cos(\theta)$ but outside $r = 2$.

    $\frac{1}{2} \int_{\alpha}^{\beta} (16\cos^2(\theta) - 4) \, d\theta$. Circle minus line area using difference formula.

29. How do you find the points of intersection of two polar curves $r = f(\theta)$ and $r = g(\theta)$?

    Solve $f(\theta) = g(\theta)$ for $\theta$. Sets equal the radial distances to find where curves meet.

30. Calculate the area between $r = 3\sin(\theta)$ and $r = 2$ from $\theta = 0$ to $\theta = \frac{\pi}{2}$.

    $\frac{1}{2} \int_{0}^{\frac{\pi}{2}} (9\sin^2(\theta) - 4) \, d\theta$. Uses the difference formula with $f(\theta) = 3\sin(\theta)$ and $g(\theta) = 2$.

31. What is the role of symmetry in simplifying polar area calculations?

    Allows reducing integration limits. Reduces computational work by exploiting curve symmetries.

32. Identify the integral bounds when finding the area between $r = 2\cos(\theta)$ and $r = 1$.

    $\theta = 0$ to $\theta = \frac{\pi}{3}$. Found by solving $2\cos(\theta) = 1$ for intersection points.

33. Choose the expression that represents the area of a single polar curve $r = f(\theta)$.

    $A = \frac{1}{2} \int_{\alpha}^{\beta} f(\theta)^2 \, d\theta$. Standard formula for area enclosed by one polar curve.

34. Find the area inside $r = 3$ but outside $r = 2\sin(\theta)$.

    $\frac{1}{2} \int_{\alpha}^{\beta} (9 - 4\sin^2(\theta)) \, d\theta$. Circle minus circle area using difference formula.

35. Identify the correct integration bounds for a full polar circle $r = a$.

    $\theta = 0$ to $\theta = 2\pi$. A complete circle requires one full rotation around the origin.

36. What is the area of the region enclosed by $r = 2 + 2\cos(\theta)$?

    $A = \frac{1}{2} \int_{0}^{2\pi} (2 + 2\cos(\theta))^2 \, d\theta$. Cardioid area formula integrated over full period.

Study these flashcards