Area between a curve and a line or two curvesEdexcel International A Level Maths: Subtopic test

1. 1
   The line $y=2x$ meets the curve $y=6x-x^2$ at the origin $O$ and at the point $A$.

   (a)

   Find the $x$-coordinate of $A$.

   [1 mark]

   • A$2$
   • B$3$
   • C$6$
   • D$4$

   (b)

   Which integral gives the area of the finite region bounded by the line and the curve?

   [1 mark]

   • A$\int_0^4(4x-x^2)\,dx$
   • B$\int_0^4(x^2-4x)\,dx$
   • C$\int_0^4(8x-x^2)\,dx$
   • D$\int_0^6(4x-x^2)\,dx$

   (c)

   Find the area of the finite region bounded by the line and the curve.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $y=x^2$ and the line $y=x+2$ intersect at the points $P$ and $Q$.

   (a)

   Find the $x$-coordinates of $P$ and $Q$.

   [1 mark]

   • A$x=1$ and $x=-2$
   • B$x=-1$ and $x=2$
   • C$x=-1$ and $x=1$
   • D$x=0$ and $x=1$

   (b)

   Find the area of the finite region bounded by the curve and the line.

   [1 mark]

   • A$-\frac92$
   • B$\frac{13}{6}$
   • C$\frac92$
   • D$\frac{10}{3}$

   (c)

   Find the area of the region bounded by the curve $y=x^2$, the $x$-axis and the lines $x=-1$ and $x=2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^2-4x+5$ and the line $l$ has equation $y=x+1$.

   (a)

   Find the coordinates of the points where $l$ meets $C$.

   [3 marks]

   (b)

   Find the area of the finite region bounded by $C$ and $l$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_1$ has equation $y=x^2$ and the curve $C_2$ has equation $y=8-x^2$. The curves intersect at two points and enclose a finite region $R$.

   (a)

   Find the coordinates of the points where $C_1$ and $C_2$ intersect, and find the exact area of $R$.

   [6 marks]

   (b)

   The line $y=4$ divides $R$ into two parts. Show that the two parts have equal area.

   [6 marks]

   Total for question 4: 12 marks

End of questions