The trapezium ruleEdexcel International A Level Maths: Subtopic test

1. 1
   The integral $I=\int_0^1\sqrt{2x+1}\,dx$ is to be estimated using the trapezium rule with four strips of equal width.

   (a)

   Find the width $h$ of each strip.

   [1 mark]

   • A$0.25$
   • B$0.2$
   • C$0.5$
   • D$4$

   (b)

   Find the value of $y=\sqrt{2x+1}$ at $x=0.5$.

   [1 mark]

   • A$1.2247$
   • B$1.4142$
   • C$1.5811$
   • D$1.7321$

   (c)

   Without calculating the estimate, state whether the trapezium rule gives an overestimate or an underestimate of $I$, giving a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The integral $\int_1^3\frac{1}{x^2}\,dx$ is estimated using the trapezium rule with four strips of equal width.

   (a)

   Which list gives the $x$-values at which $y=\frac{1}{x^2}$ is evaluated?

   [1 mark]

   • A$1,\ 2,\ 3$
   • B$1,\ 1.25,\ 1.5,\ 1.75,\ 2$
   • C$1,\ 1.5,\ 2,\ 2.5,\ 3$
   • D$1,\ 1.5,\ 2,\ 2.5$

   (b)

   Find the value of $y=\frac{1}{x^2}$ at $x=2.5$.

   [1 mark]

   • A$0.4$
   • B$0.2$
   • C$6.25$
   • D$0.16$

   (c)

   Find the exact value of $\int_1^3\frac{1}{x^2}\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The trapezium rule with four strips of equal width is used to estimate $\int_0^2 2^x\,dx$.

   (a)

   State the strip width and the five $x$-values, and calculate the corresponding values of $y=2^x$, giving values that are not exact to 4 decimal places.

   [3 marks]

   (b)

   Hence use the trapezium rule to estimate $\int_0^2 2^x\,dx$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The integral $I=\int_0^2x^3\,dx$ is to be approximated using the trapezium rule.

   (a)

   Use four strips of equal width to estimate $I$. Find the exact value of $I$ by integration, and hence calculate the percentage error in your estimate.

   [6 marks]

   (b)

   (i) Use eight strips of equal width to find a second estimate of $I$.
   (ii) Find the percentage error in this estimate and state how it compares with the error in (a).
   (iii) Explain why both estimates are greater than $I$.

   [6 marks]

   Total for question 4: 12 marks

End of questions