Definite integrals and area under a curveEdexcel A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^2-4x+3$.

   (a)

   Evaluate $\int_1^3\left(x^2-4x+3\right)\mathrm{d}x$.

   [1 mark]

   • A$\frac43$
   • B$0$
   • C$-\frac43$
   • D$-\frac{22}{3}$

   (b)

   Find the area of the finite region between $C$ and the $x$-axis for $1\le x\le3$.

   [1 mark]

   • A$-\frac43$
   • B$0$
   • C$\frac83$
   • D$\frac43$

   (c)

   Find the total area bounded by $C$, the $x$-axis and the lines $x=0$ and $x=4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=6x-x^2$ and the line $l$ has equation $y=2x$. They meet at the origin $O$ and at the point $P$.

   (a)

   Find the coordinates of $P$.

   [1 mark]

   • A$(4,\,8)$
   • B$(6,\,12)$
   • C$(2,\,4)$
   • D$(3,\,6)$

   (b)

   Evaluate $\int_0^4\left(6x-x^2\right)\mathrm{d}x$.

   [1 mark]

   • A$32$
   • B$\frac{80}{3}$
   • C$48$
   • D$\frac{64}{3}$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^3-6x^2+8x$.

   (a)

   Show that $C$ crosses the $x$-axis at $x=0$, $x=2$ and $x=4$, and find $\int y\,\mathrm{d}x$.

   [3 marks]

   (b)

   Find the total area enclosed between $C$ and the $x$-axis.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   Show that the area of the finite region bounded by $C$ and $l$ is $\frac{125}{6}$.

   [6 marks]

   (b)

   Find the total area of the finite regions bounded by $C$, the $x$-axis and the line $x=4$.

   [6 marks]

   Total for question 4: 12 marks

End of questions