7.3 Area enclosed by polar curves Quiz: Pearson Edexcel Further Maths, Unit 7

The 20 questions

1. What is the formula for the area enclosed by a polar curve r = f(θ)?

   • ½ ∫ r dθ
   • ½ ∫ r² dθ
   • ∫ r² dθ
   • 2 ∫ r² dθ

2. In polar coordinates (r, θ), what measure does the variable θ represent?

   • Radial distance
   • Direct displacement
   • Anti-clockwise angle
   • Arc length

3. What is the area enclosed by the circle r = a, where a is greater than 0?

   • a^2 / 2
   • pi a
   • pi a^2
   • 2 pi a^2

4. What is the area enclosed by the curve r = 2 for 0 <= theta <= pi/2?

   • 2
   • pi/2
   • 2 pi
   • pi

5. Find the area enclosed by r = 4 sin(theta) for 0 <= theta <= pi.

   • 16 pi
   • 2 pi
   • 8 pi
   • 4 pi

6. Find the area enclosed by r = 2 cos(theta) for -pi/2 <= theta <= pi/2.

   • pi
   • pi/2
   • 4 pi
   • 2 pi

7. Find the area enclosed by the cardioid r = a(1 + cos theta) for 0 <= theta <= 2 pi.

   • 2 pi a^2 / 3
   • pi a^2 + 2 pi a
   • 3 pi a^2 / 4
   • 3 pi a^2 / 2

8. Find the area enclosed by r = theta for 0 <= theta <= pi.

   • pi^3 / 6
   • pi^2 / 2
   • pi^3 / 2
   • pi^3 / 3

9. Find the area inside the cardioid r = 1 + cos theta but outside the circle r = 1.

   • 2 + pi/4
   • pi/4
   • 1 + pi/4
   • 2 + pi/2

10. What is the total area enclosed by the four-petal rose r = sin(2 theta)?

    • pi
    • pi/4
    • pi/8
    • pi/2

11. Tangents to a polar curve run parallel to the initial line when which derivative equals zero?

    • d²y/dθ²
    • dy/dθ
    • dr/dθ
    • dx/dθ

12. Which circular sector area formula is integrated to derive the polar area formula?

    • π r²
    • ½ r θ
    • 2 π r
    • ½ r² θ

13. Find the area enclosed by the cardioid r = 1 + sin(theta) over one full cycle.

    • 2 pi
    • 3 pi
    • pi
    • 3 pi / 2

14. Find the area of the sector of the circle r = 3 for 0 <= theta <= pi/3.

    • 3 pi / 2
    • 3 pi
    • pi/2
    • 9 pi / 2

15. Which polar equation describes an Archimedean spiral?

    • r = k theta
    • r = a(1 + cos theta)
    • r = a sec(alpha - theta)
    • r = a cos(2 theta)

16. What curve is represented by the polar equation r² = a² cos(2θ)?

    • Spiral
    • Lemniscate
    • Cardioid
    • Limaçon

17. Find the area of the sector of r = 6 between theta = pi/6 and theta = pi/2.

    • 3 pi
    • 12 pi
    • pi/3
    • 6 pi

18. Find the area enclosed by the circle r = 4 cos(theta).

    • 2 pi
    • 8 pi
    • 4 pi
    • 16 pi

19. When finding the area of a single loop of a polar curve, how many times must θ traverse it?

    • Exactly twice
    • Exactly once
    • At least once
    • Half a turn

20. Find the area enclosed by the cardioid r = 2(1 - cos theta).

    • 12 pi
    • 6 pi
    • 4 pi
    • 3 pi

Related quizzes

• Proof by mathematical induction Quiz · 1.1 · 20 questions
• Quadratic equations and complex arithmetic Quiz · 2.1-2.2 · 20 questions
• Matrix arithmetic and inverses Quiz · 3.1-3.2 · 20 questions
• Expectation of discrete random variables Quiz · 3B.1.1 · 20 questions
• The Poisson distribution Quiz · 3B.2.1 · 20 questions
• Geometric and negative binomial models Quiz · 3B.3.1 · 20 questions
• Hypothesis tests for the Poisson distribution Quiz · 3B.4.1 · 20 questions
• Applying the Central Limit Theorem Quiz · 3B.5.1 · 20 questions
• Goodness of fit tests for discrete distributions Quiz · 3B.6.1 · 20 questions
• Definition and derivation of probability generating functions Quiz · 3B.7.1 · 20 questions