Further Mathematics · Pearson Edexcel 9FM0 · Topic 5

Further calculus

Volumes made by rotating a curve about an axis, improper integrals (where a limit is infinite), the mean value of a function, and integrals that lead to inverse trigonometric functions.

7 sub-topics
1 in Year 12, 6 in Year 13
115 practice questions
7 animated lessons

Practice questions (115) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel 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.

5.1Volumes of revolution (Cartesian)Y12
5.1Volumes of revolution (parametric and modelling)Y13
5.2Improper integralsY13
5.3Mean value of a functionY13
5.4Integration using partial fractionsY13
5.5Differentiating inverse trigonometric functionsY13
5.6Integrals leading to inverse trig functionsY13

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 115 questions in this topic, free, no account needed.

Further calculus

Easy

Volume of revolution · Year 12 · 3 marks

The region under $y = x^2$ for $0 \le x \le 2$ is rotated through $2\pi$ about the $x$ -axis. Find the exact volume.
(3)

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Volume about the $x$ -axis is $\pi\int y^2\,\mathrm{d}x$ , and $y^2 = x^4$ . $V = \pi\int_0^2 x^4\,\mathrm{d}x$
Integrate. $= \pi\left[\dfrac{x^5}{5}\right]_0^2$
$= \dfrac{32\pi}{5}$

Total3

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further calculus

Not needed for Year 12Medium

An integral giving π · Year 13 · 2 marks

Evaluate $\displaystyle\int_0^1 \frac{1}{1 + x^2}\,\mathrm{d}x$ .
(2)

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$\dfrac{1}{1 + x^2}$ integrates to $\arctan x$ . $\bigl[\arctan x\bigr]_0^1$
$\arctan 1 = \tfrac{\pi}{4}$ and $\arctan 0 = 0$ . $= \dfrac{\pi}{4}$

Total2

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further calculus

Not needed for Year 12Hard

An inverse sine integral · Year 13 · 3 marks

Evaluate exactly $\displaystyle\int_0^{3/2} \frac{1}{\sqrt{9 - x^2}}\,\mathrm{d}x$ .
(3)

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$\int \dfrac{1}{\sqrt{a^2 - x^2}}\,\mathrm{d}x$ ${}= \arcsin\dfrac{x}{a}$ , with $a = 3$ . $\left[\arcsin\dfrac{x}{3}\right]_0^{3/2}$
$= \arcsin\tfrac{1}{2}$ ${}= \dfrac{\pi}{6}$

Total3

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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