Area under a curve given parametricallyEdexcel International A Level Maths: Flashcards

What these 12 flashcards ask

Area under a parametric curve?
What do t1 and t2 represent?
How do you find the limits when a curve meets the x-axis?
Why can't you use the x-limits in the t integral?
Area for x=t^2, y=4t from t=1 to t=3?
x=2t+1, y=t^2+2: what is \frac{dx}{dt}?
x=2(t-\sin t): what is \frac{dx}{dt}?
Identity used to integrate \cos^2t?
What happens if \frac{dx}{dt}<0 over the range?
Is a sketch of the curve required?
Common mistake: integrating y\,dt.
Units for an area in metres?

Exam questions on Area under a curve given parametrically

A curve has parametric equations $x=t^2$ , $y=4t$ , for $t\ge0$ . The region $R$ is bounded by the curve, the $x$ -axis and the lines $x=1$ and $x=9$ .
The lines $x=1$ and $x=9$ are replaced by the $y$ -axis and the line $x=4$ . Find the area of the new region.2 marks
A curve has parametric equations $x=2t+1$ , $y=t^2+2$ , for $t\ge0$ . The region $S$ is bounded by the curve, the $x$ -axis and the lines $x=1$ and $x=7$ .
The lines $x=1$ and $x=7$ are replaced by the lines $x=3$ and $x=5$ . Find the area of the new region.2 marks
One arch of a curve has parametric equations $x=2(t-\sin t)$ , $y=2(1-\cos t)$ for $0\le t\le2\pi$ . The region $R$ is bounded by this arch and the $x$ -axis.
Show that the area of $R$ is given by $\int_0^{2\pi}4(1-\cos t)^2\,dt$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).