Further Mathematics · AQA 7367 · Topic E

Further calculus

Improper integrals, volumes of revolution, mean values, inverse trigonometric functions, arc length, surface area and reduction formulae.

9 sub-topics
2 in Year 12, 7 in Year 13
151 practice questions
9 animated lessons

Practice questions (151) Lessons Worksheets All AQA Further Mathematics topics

What the specification says

Every sub-topic AQA 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.

E1Improper integralsY13
E2Derive and use formulae for volumes of revolutionY12
E3Mean value of a functionY12
E4Integrate using partial fractions, including quadratic factorsY13
E5Differentiate inverse trigonometric functionsY13
E6Integrals leading to inverse trigonometric functionsY13
E7Arc length and area of surface of revolution, Cartesian and parametricY13
E8Derive and use reduction formulaeY13
E9Limits of xᵏeˣ and xᵏ ln x for improper integralsY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 151 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)

Show worked solutionHide worked solution

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.

How did it go?

Report a problem with this question

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)

Show worked solutionHide worked solution

$\dfrac{1}{1 + x^2}$ 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.

How did it go?

Report a problem with this question

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)

Show worked solutionHide worked solution

$\int \dfrac{1}{\sqrt{a^2 - x^2}}\,\mathrm{d}x$ ${}= \arcsin\dfrac{x}{a}$ , 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.

How did it go?

Report a problem with this question