Further Mathematics · OCR H245 · Option A · Topic 8.06

Further calculus

Reduction formulae, arc lengths and surface areas.

  • 2 sub-topics
  • Taught in Year 13
  • 27 practice questions
  • 2 animated lessons
Practice questions (27) Lessons Worksheets coming soon All OCR Further Mathematics topics

What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. 8.06aReduction formulaeY13
  2. 8.06bArc lengths and surface areasY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 27 questions in this topic, free, no account needed.

Further calculus

Not needed for Year 12Easy

Arc length and surface area · Year 13 · 1 mark

  1. Which integral gives the length of the arc of the curve y=f(x)y = f(x)?
    (1)

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  1. A small piece of arc has length δs\delta s=1+(dy/dx)2 δx{}= \sqrt{1 + (\mathrm{d}y/\mathrm{d}x)^2}\,\delta x. Rotating it gives a band of area 2πy δs2\pi y\,\delta s.A: ∫1+(dydx)2 dx\displaystyle\int \sqrt{1 + \left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2}\,\mathrm{d}x

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Further calculus

Not needed for Year 12Medium

A reduction formula · Year 13 · 1 mark

  1. For InI_n=∫xnex dx{}= \displaystyle\int x^n\mathrm{e}^x\,\mathrm{d}x, which reduction formula does integration by parts give?
    (1)

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  1. Take u=xnu = x^n and dvdx\dfrac{\mathrm{d}v}{\mathrm{d}x}=ex{}= \mathrm{e}^x.A: In=xnex−nIn−1I_n = x^n\mathrm{e}^x - nI_{n-1}

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Further calculus

Not needed for Year 12Hard

A reduction formula · Year 13 · 7 marks

InI_n=∫01xnex dx{}= \displaystyle\int_0^1 x^n\mathrm{e}^x\,\mathrm{d}x, n≥1n \geq 1.
  1. (a)
    Show that In=e−nIn−1I_n = \mathrm{e} - nI_{n - 1}.
    (4)
  2. (b)
    Hence find the exact value of I3I_3.
    (3)
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Part (a)4 marks

  1. Integrate by parts with u=xnu = x^{n}.InI_n=[xnex]01{}= \left[x^n\mathrm{e}^x\right]_0^1−n∫01xn−1ex dx{}- n\int_0^1 x^{n - 1}\mathrm{e}^x\,\mathrm{d}x
  2. [xnex]01\left[x^n\mathrm{e}^x\right]_0^1=e{}= \mathrm{e}
  3. In=e−nIn−1I_n = \mathrm{e} - nI_{n - 1}

Part (b)3 marks

  1. I0=e−1I_0 = \mathrm{e} - 1
  2. apply the formula repeatedly\text{apply the formula repeatedly}
  3. I3=6−2eI_3 = 6 - 2 e

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