E1-E3 Improper integrals, volumes of revolution and mean value Quiz: AQA Further Maths, Unit 2

The 20 questions

1. Which feature classifies a definite integral as an improper integral?

   • Fractional power
   • Even symmetry
   • Negative integrand
   • Infinite limit

2. Which integral is improper due to an infinite limit of integration?

   • ∫ from -3 to 3
   • ∫ from 0 to 4
   • ∫ from 1 to 2
   • ∫ from 1 to ∞

3. What factor multiplies the integral from a to b to give the mean value of f(x)?

   • 1 / (a - b)
   • b - a
   • 1 / (b - a)
   • 1 / (b + a)

4. What is the formula for the volume of revolution of y about the x-axis?

   • ∫ y² dx
   • 2π ∫ y dx
   • π ∫ y² dx
   • π ∫ y dx

5. What is the value of the integral from 1 to infinity of 1/x^2 dx?

   • 2
   • infinity
   • 1/2
   • 1

6. What is the value of the integral from 0 to 1 of 1/sqrt(x) dx?

   • infinity
   • 2
   • 1/2
   • 1

7. What is the integral from 0 to 1 of 1/x dx?

   • It equals 0
   • It equals 1
   • It diverges
   • It equals ln 2

8. What is the value of the integral from 1 to infinity of 1/x^3 dx?

   • 3
   • 1
   • 1/2
   • 1/3

9. What is the value of the integral from 0 to infinity of e^(-x) dx?

   • -1
   • infinity
   • 0
   • 1

10. What is the integral from 0 to infinity of e^(-2x) dx?

    • 1
    • -1/2
    • 2
    • 1/2

11. What is the integral from 0 to 1 of ln x dx?

    • -1
    • -infinity
    • 0
    • 1

12. What is the integral from 1 to infinity of 1/(x(x + 1)) dx?

    • ln 3
    • 1
    • 1/2
    • ln 2

13. What is the mean value of 1/x on [1, e]?

    • 1/(e - 1)
    • 1/e
    • 1
    • e

14. Find the volume when y = sqrt(x) for 0 <= x <= 4 is rotated about the x-axis.

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

15. Find the volume when y = x for 0 <= x <= 2 is rotated about the x-axis.

    • 4 pi
    • 16 pi/3
    • 2 pi
    • 8 pi/3

16. Find the mean value of x^2 on [0, 3].

    • 3
    • 9/2
    • 9
    • 1

17. Find the mean value of sin x on [0, pi].

    • 0
    • 2/pi
    • pi/2
    • 1

18. Find the volume about the y-axis of the region under y = x^2 for 0 <= x <= 1, using x^2 = y.

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

19. Find the volume when the region between y = x and y = x^2 for 0 <= x <= 1 is rotated about the x-axis.

    • pi/3
    • 8 pi/15
    • 2 pi/15
    • pi/15

20. What is the value of the integral from 2 to 3 of 1/sqrt(x - 2) dx?

    • 1
    • 2
    • sqrt(2)
    • infinity

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