E7 Arc length and surface area of revolution Quiz: AQA Further Maths, Unit 2

The 20 questions

1. For a curve y = f(x), which integrand gives the arc length between x = a and x = b?

   • (dy/dx)^2
   • 1 + dy/dx
   • sqrt(1 - (dy/dx)^2)
   • sqrt(1 + (dy/dx)^2)

2. For a parametric curve x(t), y(t), which integrand gives arc length?

   • dx/dt + dy/dt
   • (dx/dt)(dy/dt)
   • sqrt((dx/dt)^2 - (dy/dt)^2)
   • sqrt((dx/dt)^2 + (dy/dt)^2)

3. Which integral gives the surface area when y = f(x) is rotated completely about the x-axis?

   • pi integral of y^2 dx
   • 2 integral of y sqrt(1 + (dy/dx)^2) dx
   • 2 pi integral of y sqrt(1 + (dy/dx)^2) dx
   • 2 pi integral of y dx

4. When the curve x = g(y) is rotated about the y-axis, which quantity is the radius used in the surface area integrand?

   • dx/dy
   • y
   • sqrt(1 + (dx/dy)^2)
   • x

5. Which integrand gives the surface area about the x-axis for parametric equations x(t), y(t)?

   • 2 pi y sqrt((dx/dt)^2 + (dy/dt)^2)
   • 2 pi y (dx/dt + dy/dt)
   • pi y^2 sqrt((dx/dt)^2 + (dy/dt)^2)
   • 2 pi x sqrt((dx/dt)^2 + (dy/dt)^2)

6. Find the arc length of y = (2/3)x^(3/2) from x = 0 to x = 3.

   • 28/3
   • 14/3
   • 7/3
   • 4

7. Find the arc length of y = ln(sec(x)) from x = 0 to x = pi/4.

   • ln(sqrt(2) - 1)
   • ln(2)
   • ln(sqrt(2) + 1)
   • 1 + sqrt(2)

8. Find the surface area when y = x from x = 0 to x = 1 is rotated completely about the x-axis.

   • pi sqrt(2)
   • pi sqrt(3)
   • pi/2
   • 2 pi sqrt(2)

9. What is the surface area of a sphere of radius r, found by rotating y = sqrt(r^2 - x^2) about the x-axis from x = -r to x = r?

   • pi r^2
   • 4 pi r^2
   • 4 pi r
   • 2 pi r^2

10. A circle is traced by x = cos(t), y = sin(t) for 0 <= t <= 2 pi. What is its arc length?

    • 4 pi
    • 2 pi
    • 1
    • pi

11. Find the arc length of y = x^2/2 from x = 0 to x = 1.

    • (1/2)(sqrt(2) - ln(1 + sqrt(2)))
    • sqrt(2) + ln(1 + sqrt(2))
    • (1/2) ln(1 + sqrt(2))
    • (1/2)(sqrt(2) + ln(1 + sqrt(2)))

12. The curve x = t^2, y = 2t for 0 <= t <= 1 is rotated completely about the x-axis. Which integral gives the surface area?

    • integral from 0 to 1 of 8 pi t sqrt(t^2 + 1) dt
    • integral from 0 to 1 of 4 pi t sqrt(t^2 + 1) dt
    • integral from 0 to 1 of 8 t sqrt(t^2 + 1) dt
    • integral from 0 to 1 of 8 pi sqrt(t^2 + 1) dt

13. Find the surface area when y = x/2 for 0 <= x <= 4 is rotated completely about the x-axis.

    • 4 pi
    • 2 sqrt(5) pi
    • 4 sqrt(5) pi
    • 8 pi

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

    • (pi/6)(17 sqrt(17) + 1)
    • (pi/6)(17 sqrt(17) - 1)
    • (pi/3)(17 sqrt(17) - 1)
    • pi(17 sqrt(17) - 1)

15. A semicircle of radius 2 is traced by x = 2 cos(t), y = 2 sin(t) for 0 <= t <= pi. What is its arc length?

    • 4 pi
    • 2 pi
    • 4
    • pi

16. Find the surface area when y = e^x for 0 <= x <= 1 is rotated completely about the x-axis.

    • integral from 0 to 1 of pi e^(2x) dx
    • integral from 0 to 1 of 2 pi e^x sqrt(1 + e^(2x)) dx
    • integral from 0 to 1 of 2 pi e^x dx
    • integral from 0 to 1 of 2 pi e^x sqrt(1 + e^x) dx

17. Which integral gives the surface area when x = cos(t), y = sin(t) for 0 <= t <= pi is rotated completely about the x-axis?

    • integral from 0 to pi of pi sin(t) dt
    • integral from 0 to pi of 2 pi sin^2(t) dt
    • integral from 0 to pi of 2 pi sin(t) dt
    • integral from 0 to pi of 2 pi cos(t) dt

18. In the surface area formula about the x-axis, what does the factor y in 2 pi y represent?

    • The width of the band in x
    • The gradient of the curve
    • The arc length of the band
    • The radius of the band, its distance from the x-axis

19. For x = t^3, y = t^2, which integrand gives the arc length element with respect to t?

    • 3 t^2 + 2 t
    • sqrt(9 t^2 + 4 t)
    • sqrt(9 t^4 + 4 t^2)
    • t^2 sqrt(9 t + 4)

20. Find the arc length of the straight line y = 2x from x = 0 to x = 3.

    • 3 sqrt(3)
    • 3 sqrt(5)
    • 5
    • 6 sqrt(5)

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