Area under a curve given parametricallyEdexcel International A Level Maths: Subtopic test

1. 1
   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 $x=9$.

   (a)

   Which integral gives the area of $R$?

   [1 mark]

   • A$\int_1^3 4t\,dt$
   • B$\int_1^9 8t^2\,dt$
   • C$\int_1^3 8t^2\,dt$
   • D$\int_1^3 4t^3\,dt$

   (b)

   Find the area of $R$.

   [1 mark]

   • A$208$
   • B$\frac{5824}{3}$
   • C$16$
   • D$\frac{208}{3}$

   (c)

   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]

   Total for question 1: 4 marks

2. 2
   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 $x=7$.

   (a)

   Which integral gives the area of $S$?

   [1 mark]

   • A$\int_0^3 2(t^2+2)\,dt$
   • B$\int_0^3 (t^2+2)\,dt$
   • C$\int_1^7 2(t^2+2)\,dt$
   • D$\int_0^3 (2t+1)\cdot2t\,dt$

   (b)

   Find the area of $S$.

   [1 mark]

   • A$15$
   • B$30$
   • C$252$
   • D$18$

   (c)

   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]

   Total for question 2: 4 marks

3. 3
   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.

   (a)

   Show that the area of $R$ is given by $\int_0^{2\pi}4(1-\cos t)^2\,dt$.

   [3 marks]

   (b)

   Hence find the exact area of $R$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A garden bed is modelled by the region bounded by the curve $C$ and the $x$-axis, with units in metres. The curve $C$ has parametric equations $x=t^2$, $y=4t-t^2$, for $0\le t\le4$.

   (a)

   Find the exact area of the garden bed.

   [6 marks]

   (b)

   A path along the line $x=4$ divides the bed into two parts. The gardener plants roses at 6 plants per square metre in the smaller part. Find the exact area of each part, and the number of rose plants needed.

   [6 marks]

   Total for question 4: 12 marks

End of questions