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Integration and Its Applications · Paper 1 — Free Maths AA HL Practice Questions

What this topic covers

Calculus is where AA HL stops rewarding memorised algebra and starts rewarding process. Every mark scheme in this topic is built around specific words — show that, hence, determine, justify — and examiners are trained to withhold marks the instant your working skips a step, even when your final answer is correct. The four subtopics here build on each other in a strict chain: limits define continuity, continuity is the gatekeeper for differentiability, derivatives feed every application (tangents, optimisation, kinematics), and integration — the reverse process — feeds differential equations. Learn it in that order and the HL-only material (implicit differentiation, integration by parts, volumes of revolution, and the whole of differential equations) stops looking like extra content and starts looking like the natural next question.

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1FoundationSAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min

A company designs a logo consisting of a curved line. The gradient of the curve at any point

(x, y)

is given by

\dfrac{dy}{dx} = 6x^2 - 4x + 5

. The curve passes through the point

(1, 8)

(a)

Find an expression for

y

in terms of

x

. [3 marks]

(b)

The designer claims the curve has no local maximum or minimum points. Justify whether this claim is correct. [2 marks]

Solutions

2MasterySAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min

A function

f

is defined for

x > 0

such that

f'(x) = \dfrac{2}{x} + 3x^2

(a)

Find

f(x)

, giving your answer in the form

f(x) = 2\ln x + x^3 + c

, where

c \in \mathbb{R}

. [2 marks]

(b)

Given that the graph of

f

passes through the point

(1, 5)

, find the value of

c

. [2 marks]

(c)

The graph of

f

has exactly one stationary point for

x > 0

. Determine the nature of this stationary point, justifying your answer. [1 mark]

Solutions

3ChallengeSAQ-LIndefinite integrals and their propertiess8 marksPaper 1~12 min

The region bounded by the curve

y = \dfrac{4}{\sqrt{x^2+4}}

, the

x

-axis, and the lines

x = 0

and

x = 2

is rotated

360°

about the

x

-axis to form a solid of revolution.

(a)

Write down a definite integral expression for the volume

V

of this solid. [1 mark]

(b)

Hence find the exact value of

V

. [5 marks]

(c)

The upper bound is extended to

x = a

. Determine whether the volume converges to a finite limit as

a \to \infty

, stating the limit if it exists. [2 marks]

Solutions

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