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

The 20 questions

1. What is an improper integral?

   • An integral with a negative lower limit only, and a bounded integrand throughout
   • An integral evaluated only by numerical methods, since no exact value exists
   • An integral whose answer is always a whole number, so it cannot be approximated
   • An integral where the integrand is undefined in the range or the range extends to infinity

2. Which integral is improper because its range extends to infinity?

   • The integral from 0 to 1 of x dx
   • The integral from -1 to 1 of x^2 dx
   • The integral from 1 to 2 of 1/x^2 dx
   • The integral from 1 to infinity of 1/x^2 dx

3. What is the mean value of f(x) on [a, b]?

   • The integral from a to b of f(x) dx alone
   • (1/(a + b)) times the integral from a to b of f(x) dx
   • (1/(b - a)) times the integral from a to b of f(x) dx
   • (b - a) times the integral from a to b of f(x) dx

4. Which formula gives the volume of revolution about the x-axis?

   • pi times the integral from a to b of x^2 dy
   • pi times the integral from a to b of y dx
   • pi times the integral from a to b of y^2 dx
   • 2 pi times the integral from a to b of 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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