Area between curvesAQA A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^2$ and the line $L$ has equation $y=2x$.

   (a)

   Find the $x$-coordinates of the points where $C$ and $L$ meet.

   [1 mark]

   • A$2$ only
   • B$0$ and $2$
   • C$-2$ and $0$
   • D$0$ and $\frac12$

   (b)

   Which expression gives the area of the region enclosed by $C$ and $L$?

   [1 mark]

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

   (c)

   Find the exact area of the region enclosed by $C$ and $L$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=x^2+1$ and the line $L$ has equation $y=x+3$.

   (a)

   Find the $x$-coordinates of the points where $C$ and $L$ meet.

   [1 mark]

   • A$-1$ and $2$
   • B$1$ and $-2$
   • C$-2$ and $2$
   • D$-1$ and $3$

   (b)

   For $x$ between the two points of intersection, which function gives the vertical distance from $C$ up to $L$?

   [1 mark]

   • A$x^2-x-2$
   • B$x^2+x+4$
   • C$2+x-x^2$
   • D$3+x-x^2$

   (c)

   Find the area of the region enclosed by $C$ and $L$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^3$ and the line $L$ has equation $y=x$.

   (a)

   Find the $x$-coordinates of the three points where $C$ and $L$ intersect.

   [3 marks]

   (b)

   Find the total area of the finite regions enclosed between $C$ and $L$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=8+2x-x^2$ and the line $L$ has equation $y=x+2$.

   (a)

   Find the exact area of the region bounded by $C$ and $L$.

   [6 marks]

   (b)

   The curve $C$ crosses the $x$-axis at two points. Find the area of the region bounded by $C$ and the $x$-axis. Hence find, as a fraction in its simplest form, the proportion of this area that lies above $L$.

   [6 marks]

   Total for question 4: 12 marks

End of questions