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

1. 1
   The curve $C$ has equation $y=3x^2-2x+1$, and $C$ lies above the $x$-axis for all values of $x$.

   (a)

   Evaluate $\int_0^2 y\,dx$.

   [1 mark]

   • A$10$
   • B$9$
   • C$6$
   • D$5$

   (b)

   Evaluate $\int_{-1}^{0} y\,dx$.

   [1 mark]

   • A$-3$
   • B$6$
   • C$1$
   • D$3$

   (c)

   Hence find the area of the region bounded by $C$, the $x$-axis and the lines $x=-1$ and $x=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=x^2-4x+3$ and crosses the $x$-axis at $A(1,0)$ and $B(3,0)$.

   (a)

   Evaluate $\int_1^3 y\,dx$.

   [1 mark]

   • A$-\frac43$
   • B$\frac43$
   • C$0$
   • D$-\frac{52}{3}$

   (b)

   Find the area of the finite region bounded by $C$ and the $x$-axis.

   [1 mark]

   • A$-\frac43$
   • B$\frac43$
   • C$\frac83$
   • D$\frac23$

   (c)

   Find the total area of the regions bounded by $C$, the $x$-axis, the $y$-axis and the line $x=3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=3\sqrt{x}+\frac{2}{x^2}$, for $x>0$.

   (a)

   Find $\int y\,dx$.

   [3 marks]

   (b)

   The finite region $R$ is bounded by $C$, the $x$-axis and the lines $x=1$ and $x=4$. Find the exact area of $R$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^3-6x^2+8x$.

   (a)

   (i) Factorise $x^3-6x^2+8x$ completely.
   (ii) Find $\int_0^2y\,dx$.

   [6 marks]

   (b)

   (i) Find $\int_2^4y\,dx$.
   (ii) Explain why this is not the area of the region bounded by $C$, the $x$-axis and the lines $x=2$ and $x=4$.
   (iii) Hence find the total area of the finite regions enclosed between $C$ and the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions