What does the further calculus strand in OCR Further Maths A cover?

It covers the Maclaurin series (the general formula, the standard expansions for e^x, ln(1+x), sin x and cos x, finding a series by repeated differentiation or by combining known series, and approximation), improper integrals (with an infinite limit or an unbounded integrand, evaluated as a limit, and deciding convergence), volumes of revolution (about the x-axis and y-axis, between two curves, and parametric and improper cases), and differential equations (first order linear by the integrating factor, second order constant-coefficient by the auxiliary equation and particular integral, and simple harmonic motion and damping). It is part of the mandatory Pure Core.

Which further calculus topics carry the most marks in OCR Further Maths A?

Differential equations are the heaviest, because first and second order solutions, with initial conditions and damping classification, appear as long structured questions. Volumes of revolution and the Maclaurin series are reliable high-tariff questions, and improper integrals appear both alone and inside volume questions. Fluency with the integrating factor and the auxiliary equation is the single biggest mark-winner in this strand.

What techniques recur across the OCR further calculus content?

Quoting the standard Maclaurin series and substituting or combining them, building a series by repeated differentiation, replacing an infinite or awkward limit by a variable and taking a limit, integrating pi y squared (or pi x squared) for a volume and subtracting squares between two curves, solving first order linear equations with an integrating factor, and solving second order equations through the auxiliary equation plus a particular integral. These are automatic by exam day.

What is the most common further calculus mistake in OCR Further Maths A?

Forgetting to square y in a volume of revolution, and substituting an infinite limit directly instead of taking a limit. Other frequent errors are dropping the factorials in a Maclaurin series, using a series outside its range of validity, subtracting curves before squaring them between two curves, choosing the wrong integrating factor, and forgetting the particular integral for a non-zero right-hand side. Always take limits explicitly and show full working.

How should I revise the OCR further calculus content?

Work topic by topic against the specification, drilling each technique until it is automatic, then practise mixed problems under timed conditions. Keep a sheet of standard results: the four Maclaurin series, the integrating factor formula, the auxiliary-equation cases, and the volume formulae. Always show the limit step for an improper integral and the complementary function plus particular integral for a second order equation, and classify damping from the roots.

EnglandFurther Maths

OCR A-Level Further Maths A Further calculus: Maclaurin series, improper integrals, volumes and differential equations

A deep-dive OCR A-Level Further Mathematics A guide to the further calculus strand of the Pure Core: the Maclaurin series, improper integrals and convergence, volumes of revolution, and first and second order differential equations including simple harmonic motion and damping, with the techniques OCR repeats in the Pure Core papers.

Generated by Claude Opus 4.818 min readH245Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What the further calculus strand demands
Series and improper integrals
Volumes and differential equations
How the further calculus content is examined
Check your knowledge

What the further calculus strand demands

The further calculus strand of OCR A-Level Further Mathematics A (H245) is part of the mandatory Pure Core, assessed across both Pure Core papers (Y540 and Y541). It extends A-Level Mathematics calculus with series, limiting processes and differential equations. The examiners reward fluent technique, explicit limit steps, and clear structure in differential-equation solutions.

This guide walks through the four further calculus topics in a logical order, then sets out the exam patterns OCR repeats. Each topic has a matching dot-point page with worked exam questions; this overview ties them together.

Series and improper integrals

Maclaurin series expands a function in powers of $x$ about zero, either by repeated differentiation and the general formula or by substituting into and combining the standard series for $e^x$ , $\ln(1 + x)$ , $\sin x$ and $\cos x$ , and truncating to approximate. Improper integrals handle an infinite limit of integration or an unbounded integrand by replacing the awkward limit with a variable and taking a limit, then deciding whether the integral converges (finite limit) or diverges.

Volumes and differential equations

Volumes of revolution find the volume swept when a region rotates about an axis: $\pi\int y^2\,dx$ about the $x$ -axis, $\pi\int x^2\,dy$ about the $y$ -axis, and a difference of squares between two curves, with parametric and improper variants. Differential equations solve first order linear equations by the integrating factor and second order constant-coefficient equations by the auxiliary equation (complementary function) plus a particular integral, and interpret them as simple harmonic motion and damped systems.

How the further calculus content is examined

A typical OCR profile for this strand:

Series questions. Find a Maclaurin series and use it to approximate a value.
Improper integral questions. Evaluate an integral with an infinite limit, or decide convergence.
Volume questions. Rotate a region about an axis, or find the volume between two curves.
Differential equation questions. Solve a first or second order equation with conditions, and classify the damping.

Check your knowledge

A mix of recall and technique questions covering the further calculus content. Attempt them under timed conditions, then check against the solutions.

State the Maclaurin series of $e^x$ up to the term in $x^2$ . (1 mark)
Write $\displaystyle\int_1^{\infty} x^{-2}\,dx$ as a limit. (1 mark)
State the formula for the volume when a region under $y = f(x)$ is rotated about the $x$ -axis. (1 mark)
State the integrating factor for $\dfrac{dy}{dx} + 5y = x$ . (1 mark)
Form the auxiliary equation for $\ddot{x} + 3\dot{x} + 2x = 0$ . (1 mark)
What type of damping corresponds to complex auxiliary roots? (1 mark)

Check your knowledge: further calculus

Recall questions covering the key formulas and techniques from the further calculus strand.

Step 1: State the Maclaurin series of $e^x$ up to the term in $x^2$

The Maclaurin series expands $e^x$ as an infinite sum of powers of $x$ divided by factorials; truncating after the $x^2$ term gives the standard quadratic approximation.

Answer: $e^x \approx 1 + x + \dfrac{x^2}{2}$ .

Step 2: Write $\displaystyle\int_1^{\infty} x^{-2}\,dx$ as a limit

An improper integral with an infinite upper limit is defined by replacing infinity with a finite variable and taking the limit as that variable grows without bound.

Answer: $\displaystyle\lim_{t \to \infty}\int_1^{t} x^{-2}\,dx$ .

Step 3: State the formula for the volume when a region under $y = f(x)$ is rotated about the $x$ -axis

Rotating a thin vertical strip of width $dx$ and height $y$ about the $x$ -axis generates a disc of volume $\pi y^2\,dx$ ; integrating over the interval gives the total volume.

Answer: $V = \pi\displaystyle\int_a^b y^2\,dx$ .

Step 4: State the integrating factor for $\dfrac{dy}{dx} + 5y = x$

For a first order linear equation $\dfrac{dy}{dx} + P(x)y = Q(x)$ , the integrating factor is $e^{\int P\,dx}$ . Here $P = 5$ , so the integrating factor is $e^{5x}$ .

Answer: $I = e^{\int 5\,dx} = e^{5x}$ .

Step 5: Form the auxiliary equation for $\ddot{x} + 3\dot{x} + 2x = 0$

To solve a second order constant-coefficient ODE, substitute $x = e^{mt}$ to convert derivatives into powers of $m$ ; each dot contributes one factor of $m$ .

Answer: $m^2 + 3m + 2 = 0$ (which factorises as $(m + 1)(m + 2) = 0$ ).

Step 6: State what type of damping corresponds to complex auxiliary roots

When the auxiliary equation has complex conjugate roots, the complementary function contains sine and cosine factors multiplied by a decaying exponential, so the system oscillates with amplitude that decreases over time.

Answer: Light (under) damping: the system oscillates with decaying amplitude.

Sources & how we know this

OCR A Level Further Mathematics A (H245) specification — OCR (2017)