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Integration and Its Applications · Paper 2 — 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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1MasterySAQ-SIndefinite integrals and their propertiess7 marksPaper 2~11 min

A particle moves along the

x

-axis. Its velocity at time

t

seconds is given by

v(t) = t^2 \cos(2t)\,\text{m s}^{-1}

for

t \geq 0

(a)

Show that

\displaystyle\int t^2 \cos(2t)\,dt = \frac{1}{2}t^2\sin(2t) + \frac{1}{2}t\cos(2t) - \frac{1}{4}\sin(2t) + C

. [3 marks]

(b)

Hence, find the displacement of the particle from

t = 0

t = \dfrac{\pi}{2}

, giving your answer in exact form. [2 marks]

(c)

Determine the total distance travelled by the particle from

t = 0

t = \dfrac{\pi}{2}

. Give your answer correct to 3 significant figures. [2 marks]

Solutions

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