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

The 20 questions

1. Which formula gives the area enclosed by the polar curve r = f(theta) between theta = alpha and theta = beta?

   • Area = integral from alpha to beta of r d(theta)
   • Area = (1/2) integral from alpha to beta of r^2 d(theta)
   • Area = (1/2) integral from alpha to beta of r d(theta)
   • Area = integral from alpha to beta of r^2 d(theta)

2. What does theta represent in polar coordinates (r, theta)?

   • The distance from the pole to the point
   • The gradient of the tangent at the point
   • The angle measured from the initial line
   • The perpendicular height above the initial line

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. At which points does the tangent to r = f(theta) run parallel to the initial line?

    • Where r = 0 on the curve
    • Where dy/d(theta) = 0 and dx/d(theta) is not zero
    • Where dr/d(theta) = 0 only
    • Where dx/d(theta) = 0 and dy/d(theta) is not zero

12. Why does the polar area formula use r^2 rather than r?

    • Because the polar area always equals the integral of r with respect to theta
    • Because the arc length of a polar curve is the integral of r^2
    • Because theta must be converted to degrees before r is squared
    • Because a thin sector of radius r and angle d(theta) has area (1/2) r^2 d(theta)

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. Which shape is described by r^2 = a^2 cos(2 theta)?

    • A circle centred on the pole with radius a
    • A lemniscate (figure of eight)
    • A cardioid with a single inner loop
    • An Archimedean spiral that winds outward

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, what must the chosen theta-interval do?

    • Cover the full interval from 0 to 2 pi
    • Trace that loop exactly once
    • Give the largest possible value of r
    • Make r equal to zero at both ends

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

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

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