A-level · Integration

Areas under parametric curvesExam-style questions

Finding the area under a curve given by parametric equations. Why the area is the integral of y times dx/dt with respect to t; turning the x-limits into t-limits; the region a curve encloses with the x-axis; which way round the limits go when x decreases as t increases; curves with trig parameters; and regions with a straight edge, made of a curved piece and a triangle.

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24 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
The diagram shows the curve with parametric equations $x = t^2,\ y = 4t - t^2$ , which meets the $x$ -axis at $t = 0$ and $t = 4$ . Find the area of the shaded region enclosed by the curve and the $x$ -axis.
Level 4 · 3 marksTry it
2
A curve has parametric equations $x = t^2 + 1,\ y = t^3$ , $t \ge 0$ . Find the area of the region bounded by the curve, the $x$ -axis and the lines $x = 1$ and $x = 5$ .
Level 4 · 3 marksTry it
3
A curve has parametric equations $x = t^{3}$ , $y = t(2 - t)$ , for $t \geqslant 0$ . The curve meets the $x$ -axis where $t = 0$ and $t = 2$ .

Find the area of the region enclosed by the curve and the $x$ -axis. (5 marks)
Level 4 · 5 marksTry it
4
The curve $C$ has parametric equations

$x = t^3 + 2 \qquad\qquad y = 9 - t^2 \qquad\qquad -3 \leqslant t \leqslant 3$
(a) Find the coordinates of the two points where $C$ meets the $x$ -axis.

(2 marks)

The finite region $R$ is bounded by $C$ and the $x$ -axis.
(b) Using algebraic integration, find the exact area of $R$ .

(4 marks)
Level 4 · 6 marksTry it
5
The curve $C$ has parametric equations

$x = 4\tan t \qquad\qquad y = 5\sin 2t \qquad\qquad 0 \leqslant t < \frac{\pi}{2}$

The point $M$ is the maximum point of $C$ . The finite region $R$ is bounded by $C$ , the $x$ -axis and the line through $M$ parallel to the $y$ -axis.
(a) Show that the area of $R$ is given by

$\int_{0}^{a} 40\tan t\,dt$

where $a$ is a constant to be found.

(5 marks)
(b) Hence find the area of $R$ , giving your answer to 3 significant figures.

(1 mark)
Level 4 · 6 marksTry it

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