G3 Area enclosed by a polar curve Quiz: AQA Further Maths, Unit 2

The 20 questions

1. The area enclosed by a polar curve r = f(theta) between theta = a and theta = b is given by which formula?

   • integral from a to b of r d(theta)
   • (1/2) integral from a to b of r d(theta)
   • (1/2) integral from a to b of r^2 d(theta)
   • integral from a to b of r^2 d(theta)

2. What is the area enclosed by the circle r = a?

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

3. Find the area enclosed by r = 2 for 0 <= theta <= pi/2.

   • pi/2
   • 2 pi
   • 4 pi
   • pi

4. What is the area enclosed by the full cardioid r = 1 + cos(theta)?

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

5. What is the area enclosed by the full curve r = 2 cos(theta)?

   • 2 pi
   • pi/2
   • pi
   • 4 pi

6. Find the area enclosed by r = 3 for 0 <= theta <= pi/3.

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

7. What is the area of one petal of the rose r = sin(2 theta)?

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

8. What is the area enclosed by the full curve r = 2(1 - cos(theta))?

   • 6 pi
   • 2 pi
   • 12 pi
   • 3 pi

9. Find the area enclosed by the full curve r = 4 sin(theta).

   • 2 pi
   • pi
   • 8 pi
   • 4 pi

10. Which integral gives the area enclosed by the full curve r = 2 + cos(theta)?

    • (1/2) integral from 0 to 2 pi of (2 + cos(theta))^2 d(theta)
    • (1/2) integral from 0 to 2 pi of (2 + cos^2(theta)) d(theta)
    • integral from 0 to 2 pi of (2 + cos(theta))^2 d(theta)
    • (1/2) integral from 0 to 2 pi of (2 + cos(theta)) d(theta)

11. Why is the polar area element (1/2) r^2 d(theta) rather than r d(theta)?

    • Because theta is measured in degrees
    • Because area is always linear in r
    • Because r is always equal to 1 on any curve
    • A thin sector has area approximately (1/2) r^2 d(theta), which is quadratic in r

12. Which limits of theta give one complete traversal of the circle r = a?

    • 0 to pi/2
    • -pi/2 to pi/2
    • 0 to pi
    • 0 to 2 pi

13. Find the area enclosed by r = 3 sin(theta).

    • 9 pi/4
    • 3 pi/4
    • 9 pi/2
    • 3 pi

14. Find the area enclosed by r = 1 + cos(theta) for 0 <= theta <= pi.

    • 3 pi/2
    • pi/2
    • pi
    • 3 pi/4

15. Find the area enclosed by r = 4 for 0 <= theta <= pi.

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

16. Find the area enclosed by r = 2 for pi/4 <= theta <= pi/2.

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

17. Find the area of the region between the curves r = 2 and r = 4 for 0 <= theta <= pi/2.

    • 3 pi
    • 6 pi
    • 12 pi
    • pi

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

    • 2 pi/3 + sqrt(3)/2
    • pi/3 + sqrt(3)/4
    • pi/6 + sqrt(3)/8
    • pi/3 - sqrt(3)/4

19. A sector of the circle r = 5 spans theta from 0 to 2 radians. What is its area?

    • 5
    • 12.5
    • 50
    • 25

20. What is the area enclosed by one loop of the curve r = cos(3 theta)?

    • pi/24
    • pi/12
    • pi/4
    • pi/6

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