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 $x = t^2,\ y = t$ . The shaded region is bounded by the curve, the $x$ -axis and the line $x = 9$ . Find its area.
Level 3 · 3 marksTry it
2
A curve has parametric equations $x = 2t,\ y = t^2 + 1$ . The shaded region is bounded by the curve, the $x$ -axis and the lines $x = 0$ and $x = 4$ . Find its area.
Level 3 · 3 marksTry it
3
A curve has parametric equations $x = 3t$ , $y = t^{2}$ . Find the area of the region bounded by the curve, the $x$ -axis, and the lines corresponding to $t = 0$ and $t = 2$ . (4 marks)
Level 3 · 4 marksTry it
4
A curve has parametric equations $x = t^{2}$ , $y = 2t + 1$ , for $t \geqslant 0$ . Find the area of the region bounded by the curve, the $x$ -axis, and the lines corresponding to $t = 1$ and $t = 3$ . (5 marks)
Level 3 · 5 marksTry it
5
A curve has parametric equations $x = t + 1$ , $y = 4 - t^{2}$ . The curve meets the $x$ -axis where $t = -2$ and $t = 2$ .

Find the area of the region enclosed by the curve and the $x$ -axis. (5 marks)
Level 3 · 5 marksTry it

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