8.06a-b Reduction formulae, arc lengths and surface areas of revolution Quiz: OCR Further Maths, Unit 5

The 20 questions

1. What standard integration technique is primarily used to derive reduction formulae for definite integrals?

   • Substitution
   • Integration by parts
   • Implicit differentiation
   • Partial fractions

2. What is the integrand in the Cartesian formula for arc length with respect to x?

   • √(1 + (dy/dx)²)
   • √(1 + dy/dx)
   • √(1 - (dy/dx)²)
   • 1 + (dy/dx)²

3. What is the formula for calculating the arc length of a parametric curve?

   • ∫ √((dx/dt)² + (dy/dt)²) dt
   • ∫ (dx/dt + dy/dt) dt
   • ∫ ((dx/dt)² + (dy/dt)²) dt
   • ∫ √((dx/dt) + (dy/dt)) dt

4. What is the integrand in the polar formula for arc length with respect to θ?

   • r² + (dr/dθ)²
   • √(1 + (dr/dθ)²)
   • √(r² + (dr/dθ)²)
   • √(r + dr/dθ)

5. What constant factor precedes the integral sign in the surface area of revolution formula?

   • π
   • 1/2 π
   • 4π
   • 2π

6. What expression replaces y when rotating a polar curve about the initial line?

   • r² sin θ
   • r sin θ
   • r tan θ
   • r cos θ

7. For the integral In = ∫ x^n e^x dx, which reduction formula is correct?

   • In = x^n e^x - n I_{n-1}
   • In = n x^n e^x - I_{n-1}
   • In = x^n e^x - I_{n-1}
   • In = x^n e^x + n I_{n-1}

8. For the reduction integral In = ∫ (ln x)^n dx, what is the value of I0?

   • 0
   • 1 + C
   • ln x + C
   • x + C

9. What is the arc length of the straight line y = 2x from x = 0 to x = 3?

   • 3√5
   • 3√3
   • 3√2
   • 6

10. For parametric equations x = t² and y = t³, what is the arc length integrand dt?

    • √(2t + 3t²)
    • √(4t² + 9t⁴)
    • 2t + 3t²
    • √(4t + 9t²)

11. For the standard integral In = ∫₀^(π/2) sin^n x dx, what is the value of I0?

    • π/2
    • π
    • 0
    • 1

12. For the standard integral In = ∫₀^(π/2) sin^n x dx, what is the value of I1?

    • 1
    • 2
    • π/2
    • 0

13. Which formula calculates the surface area of revolution when y = f(x) rotates about the x-axis?

    • π ∫ y² dx
    • 2π ∫ x √(1 + (dy/dx)²) dx
    • 2π ∫ y √(1 + (dy/dx)²) dx
    • π ∫ y √(1 + (dy/dx)²) dx

14. What is the surface area when y = x from x = 0 to x = 1 rotates about the x-axis?

    • π/2
    • 2π
    • π√2
    • 2π√2

15. Using the polar arc length formula, what is the perimeter of the circle r = a?

    • πa²
    • πa
    • 4πa
    • 2πa

16. Given In = e - n I_{n-1} for In = ∫₀¹ x^n e^x dx and I0 = e - 1, what is I1?

    • e - 1
    • e
    • e - 2
    • 1

17. For In = ∫₀^(π/2) cos^n x dx, which reduction formula links In to I_{n-2}?

    • In = ((n-1)/n) I_{n-2}
    • In = (1/n) I_{n-2}
    • In = ((n-2)/n) I_{n-2}
    • In = (n/(n-1)) I_{n-2}

18. What is the surface area formed by rotating y = 3 from x = 0 to x = 2 about x-axis?

    • 36π
    • 12π
    • 6π
    • 18π

19. What is the arc length of x = cos t, y = sin t from t = 0 to t = π?

    • 1
    • 2π
    • π/2
    • π

20. What is the surface area when x = t, y = 2t from t = 0 to 1 rotates around the y-axis?

    • 2π
    • π√5
    • 4π
    • 2π√5

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