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

1. 1
   The curve $C_1$ has equation $y=3x^2-2x+4$.

   (a)

   Evaluate $\int_1^3\left(3x^2-2x+4\right)dx$.

   [1 mark]

   • A$-26$
   • B$30$
   • C$34$
   • D$26$

   (b)

   Evaluate $\int_3^1\left(3x^2-2x+4\right)dx$.

   [1 mark]

   • A$-26$
   • B$26$
   • C$-30$
   • D$-34$

   (c)

   Explain why $\int_1^3\left(3x^2-2x+4\right)dx$ is equal to the area of the region bounded by $C_1$, the $x$-axis and the lines $x=1$ and $x=3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C_2$ has equation $y=(x-1)(x-5)$. The finite region $R$ is bounded by $C_2$ and the $x$-axis.

   (a)

   Evaluate $\int_1^5(x-1)(x-5)\,dx$.

   [1 mark]

   • A$\frac{32}{3}$
   • B$-\frac{32}{3}$
   • C$-\frac{25}{3}$
   • D$-6$

   (b)

   Find the area of $R$.

   [1 mark]

   • A$-\frac{32}{3}$
   • B$\frac{25}{3}$
   • C$\frac{32}{3}$
   • D$16$

   (c)

   Find the area of the finite region bounded by $C_2$, the coordinate axes and the line $x=1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C_3$ has equation $y=4x-x^3$.

   (a)

   Find the coordinates of the points where $C_3$ meets the $x$-axis.

   [3 marks]

   (b)

   Find the total area of the finite regions bounded by $C_3$ and the $x$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_4$ has equation $y=6x-x^2$. The finite region $R$ is bounded by $C_4$ and the $x$-axis.

   (a)

   (i) Find the coordinates of the points where $C_4$ meets the $x$-axis.
   (ii) Find the area of $R$.

   [6 marks]

   (b)

   (i) Evaluate $\int_0^8\left(6x-x^2\right)dx$.
   (ii) Explain why this is not the area bounded by $C_4$, the $x$-axis and the lines $x=0$ and $x=8$, and find that area.

   [6 marks]

   Total for question 4: 12 marks

End of questions