15. Applications of Integration

K341 prints a MATHEMATICAL FORMULAE page at the front of every A-Math paper. Everything below is given to you, so no marks go for recalling it — they go for knowing which one the question wants.

Algebra

Quadratic equationfor ax² + bx + c = 0	$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ The E-Math paper does not give you this one.
Binomial expansion	${(a + b)}^{n} = a^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a^{n - 1} b + \dots + (\begin{matrix} n \\ r \end{matrix}) a^{n - r} b^{r} + \dots + b^{n}$ for positive integer n
and the coefficient	$(\begin{matrix} n \\ r \end{matrix}) = \frac{n!}{r! (n - r)!}$

Trigonometry — identities

Pythagorean	$\sin^{2} A + \cos^{2} A = 1$ $\sec^{2} A = 1 + \tan^{2} A$ ${cosec}^{2} A = 1 + \cot^{2} A$
Compound angle	$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Double angle	$\sin 2 A = 2 \sin A \cos A$ $\cos 2 A = \cos^{2} A - \sin^{2} A = 2 \cos^{2} A - 1 = 1 - 2 \sin^{2} A$ $\tan 2 A = \frac{2 \tan A}{1 - \tan^{2} A}$

Trigonometry — formulae for △ABC

Sine Rule	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine Rule	$a^{2} = b^{2} + c^{2} - 2 b c \cos A$
Area	$△ = \frac{1}{2} b c \sin A$

None of this is printed on your paper. K341's sheet is generous with trigonometry and silent about almost everything else — including one formula it asks you to use by name.

Quadratic functions am-c1

Completed square form	$y = a {(x - h)}^{2} + k$ turning point (h, k); minimum if a > 0, maximum if a < 0
Always positive / always negative	a > 0 and the minimum value > 0 — or a < 0 and the maximum < 0 equivalently a > 0 with $b^{2} - 4 a c < 0$

The R-formula — asked for by name, and not given am-c10

Harmonic form

a \cos θ + b \sin θ = R \cos (θ \pm α)

with

R = \sqrt{a^{2} + b^{2}}

and

\tan α = \frac{b}{a}

— K341 lists using it as a G1 bullet, then does not print it.

Logarithms and exponentials am-c6

The three laws	$\log a (x y) = \log a x + \log a y$ and the quotient and power laws
Equivalence	$a^{x} = y$ ⇔ $\log a y = x$
Change of base	$\log a b = \frac{\log c b}{\log c a}$

Algebra the sheet skips am-c3 am-c4

Sum and difference of cubes	$a^{3} \pm b^{3} = (a \pm b) (a^{2} \mp a b + b^{2})$
Remainder and factor theorems	the remainder on dividing by (x − a) is f(a); it is a factor exactly when f(a) = 0
Surds	rationalising a denominator, and the conjugate $a - b \sqrt{k}$

Coordinate geometry am-c7 am-c8

Parallel and perpendicular	$m_{1} = m_{2}$ and $m_{1} m_{2} = - 1$
Midpoint	$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
Circle	${(x - a)}^{2} + {(y - b)}^{2} = r^{2}$ and the expanded form with 2g, 2f, c
Area of a rectilinear figure	the shoelace array, taken anticlockwise
Linear law	rewrite $y = a x^{n}$ or $y = k b^{x}$ as $Y = m X + c$ by taking logs

The two papers (K341 G3 Additional Mathematics)

Paper 1 · 2 h 15 min · 90 marks · 50%	12 to 14 questions of varying marks and lengths, up to 10 marks each. Answer all of them.
Paper 2 · 2 h 15 min · 90 marks · 50%	9 to 11 questions of varying marks and lengths, up to 12 marks each. Answer all of them.

The G3 Mathematics syllabus is assumed: it is not tested directly, but a question may lean on it.

What the front page of every paper says

These are the rules printed on the paper itself. Quoted from the syllabus; the words in brackets are ours.

Working	“Omission of essential working will result in loss of marks.” (A bare answer earns the answer mark at most; the method marks sit on the lines that lead to it.)
Accuracy	“Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. In questions which explicitly require an answer to be shown to be correct to a specific accuracy, the answer must be first shown to a higher degree of accuracy.” (Round once, at the end. Carry at least 4 s.f. through the working, or keep the value in the calculator.)
π	“Unless the question requires the answer in terms of π, the calculator value for π or π = 3.142 should be used.”
Units and time	SI units are used for mass and measures; compound units use the solidus (5 m/s, 13.6 g/cm³); the 12-hour and 24-hour clock may both be used (3.15 p.m. is 15 15).
Calculators	“An approved calculator may be used in both Paper 1 and Paper 2.” (A question that says without using a calculator or do not use a calculator is marked on the exact working it asks for.)
Formulae	“Relevant mathematical formulae will be provided for candidates.” (Exactly what is provided, and what is not, is the other two tabs of this window.)

What this means when you write

Exact answers stay exact: a fraction, a surd, a multiple of π. Only a decimal that does not terminate gets rounded, to 3 s.f., and only on the answer line. Angles in degrees go to 1 d.p.; money to the nearest cent.

Show the line that chooses the method (the formula written out, the equation set up, the rule named) before the line that uses it. That is where the method mark lives.