Simpson's ruleEdexcel A-Level Further Maths: Subtopic test

1. 1
   Simpson's rule with 4 strips is used to estimate $\int_{0}^{2}f(x)\,dx$. The values of $f(x)$ at $x=0,\ 0.5,\ 1,\ 1.5,\ 2$ are $1,\ 3,\ 4,\ 6,\ 7$ respectively.

   (a)

   What is the width $h$ of each strip?

   [1 mark]

   • A$0.25$
   • B$2$
   • C$0.5$
   • D$4$

   (b)

   Which value is the estimate of the integral?

   [1 mark]

   • A$\frac{26}{3}$
   • B$\frac{17}{2}$
   • C$7$
   • D$26$

   (c)

   The exact value of the integral is $8.7$. Find the percentage error in the Simpson's rule estimate, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Simpson's rule is used with 4 strips to estimate $\int_{2}^{10}f(x)\,dx$. The values of $f(x)$ at $x=2,\ 4,\ 6,\ 8,\ 10$ are $3,\ 5,\ 8,\ 12,\ 17$ respectively.

   (a)

   What is the width $h$ of each strip?

   [1 mark]

   • A$4$
   • B$2$
   • C$1$
   • D$\frac{8}{5}$

   (b)

   Which ordinates are multiplied by 4 in Simpson's rule?

   [1 mark]

   • A$3$ and $17$
   • B$8$ only
   • C$5,\ 8$ and $12$
   • D$5$ and $12$

   (c)

   Use the values to find the Simpson's rule estimate of the integral, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_{0}^{1}\frac{4}{1+x^{2}}\,dx$, whose exact value is $\pi$. A calculator may be used.

   (a)

   Use Simpson's rule with 4 strips to estimate $I$, giving your answer to 5 significant figures.

   [3 marks]

   (b)

   Use Simpson's rule with 2 strips to estimate $I$. By finding the error in each estimate, state with a reason which of the two estimates is more accurate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $I=\int_{1}^{5}\frac{1}{x}\,dx$. A calculator may be used.

   (a)

   Use Simpson's rule with 4 strips to estimate $I$, and find the percentage error in your estimate, giving your answers to 4 significant figures.

   [6 marks]

   (b)

   Use Simpson's rule with 8 strips to estimate $I$ and find the percentage error to 4 significant figures. Compare the error with your answer to part (a) and explain the change.

   [6 marks]

   Total for question 4: 12 marks

End of questions