Mid-ordinate and Simpson's rulesAQA A-Level Further Maths: Subtopic test

1. 1
   The integral $I=\int_0^1\frac{1}{1+x}\,dx$ is to be estimated using four strips of equal width. Let $y_r=\frac{1}{1+0.25r}$.

   (a)

   At which values of $x$ does the mid-ordinate rule with four strips evaluate the function?

   [1 mark]

   • A$0,\ 0.25,\ 0.5,\ 0.75$
   • B$0.25,\ 0.5,\ 0.75,\ 1$
   • C$0,\ 0.25,\ 0.5,\ 0.75,\ 1$
   • D$0.125,\ 0.375,\ 0.625,\ 0.875$

   (b)

   Which expression is Simpson's rule estimate of $I$ using these four strips?

   [1 mark]

   • A$\frac{0.25}{3}[y_0+y_4+2(y_1+y_3)+4y_2]$
   • B$0.25[y_0+4y_1+2y_2+4y_3+y_4]$
   • C$\frac{0.25}{3}[y_0+y_4+4(y_1+y_3)+2y_2]$
   • D$\frac{0.25}{2}[y_0+y_4+2(y_1+y_2+y_3)]$

   (c)

   Use the mid-ordinate rule with four strips to estimate $I$, to 4 decimal places.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The depth of a river is measured at 1 m intervals across its 6 m width. The depths, in metres, from one bank to the other are $0,\ 1.2,\ 2.0,\ 2.4,\ 1.8,\ 1.0,\ 0$. The cross-sectional area is the area under the depth profile.

   (a)

   Use Simpson's rule with six strips to estimate the cross-sectional area.

   [1 mark]

   • A$8.67$ m$^2$
   • B$8.4$ m$^2$
   • C$26$ m$^2$
   • D$8.13$ m$^2$

   (b)

   Why can Simpson's rule be applied to these measurements?

   [1 mark]

   • AThe strips have width 1 m
   • BThere is an even number of strips
   • CThe first and last depths are zero
   • DAll the depths are positive

   (c)

   Use the mid-ordinate rule with three strips of width 2 m to estimate the cross-sectional area.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider $I=\int_0^2x^3\,dx$.

   (a)

   Use Simpson's rule with two strips to estimate $I$, and show that it equals the exact value of $I$.

   [3 marks]

   (b)

   Use the mid-ordinate rule with two strips to estimate $I$. Say whether this is an underestimate or an overestimate, and justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The integral $I=\int_0^1\sqrt{1+x^3}\,dx$ is to be estimated using four strips of equal width. A calculator may be used. Give estimates to 5 decimal places.

   (a)

   Find the mid-ordinate rule estimate and Simpson's rule estimate of $I$ using four strips.

   [6 marks]

   (b)

   The exact value of $I$ is $1.11145$ to 5 decimal places.
   (i) Find the percentage error of each of your estimates.
   (ii) Explain why Simpson's rule is more accurate here.
   (iii) State one change that would improve both estimates.

   [6 marks]

   Total for question 4: 12 marks

End of questions