The trapezium ruleAQA A-Level Maths: Subtopic test

1. 1
   The integral $I=\int_0^2x^2\,dx$ is estimated using the trapezium rule with 4 strips of equal width.

   (a)

   Find the trapezium rule estimate of $I$.

   [1 mark]

   • A$1.875$
   • B$5.5$
   • C$2.75$
   • D$2.67$

   (b)

   Which statement about the estimate is correct?

   [1 mark]

   • AIt is an underestimate, because the chords lie below the curve.
   • BIt is an overestimate, because $x^2$ is convex so the chords lie above the curve.
   • CIt is exact, because four strips are enough for a quadratic.
   • DIts size cannot be determined without the exact value of $I$.

   (c)

   Find the exact value of $I$ and the percentage error in the estimate, to 2 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A river channel is 6 m wide. The depth of the water is measured every 1 m across the channel, giving 0, 0.8, 1.5, 1.9, 1.6, 0.9 and 0 metres. The trapezium rule is used to estimate the area of the cross-section of the river.

   (a)

   How many strips are used, and how wide is each strip?

   [1 mark]

   • A5 strips, each 1.2 m wide
   • B7 strips, each 1 m wide
   • C12 strips, each 0.5 m wide
   • D6 strips, each 1 m wide

   (b)

   Find the trapezium rule estimate for the area of the cross-section.

   [1 mark]

   • A$6.7$ m$^2$
   • B$3.35$ m$^2$
   • C$13.4$ m$^2$
   • D$6.0$ m$^2$

   (c)

   The water flows through the channel at an average speed of $0.4$ m s$^{-1}$. Estimate the volume of water that passes through the cross-section each second.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The integral $J=\int_1^2\frac1x\,dx$ is estimated using the trapezium rule with 4 strips of equal width.

   (a)

   Find the trapezium rule estimate of $J$, giving your answer to 4 significant figures.

   [3 marks]

   (b)

   The exact value of $J$ is $\ln2=0.6931$ (4 d.p.). State, with a reason, whether your estimate is an overestimate or an underestimate, and find the percentage error in the estimate, to 2 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f(x)=2^x$ is used to find the area under the curve between $x=0$ and $x=4$ with 4 strips of width 1. The exact value of $\int_0^42^x\,dx$ is $\frac{15}{\ln2}=21.64$ (4 s.f.).

   (a)

   (i) Find a lower bound and an upper bound for the area using rectangles of width 1. (ii) Explain why these are bounds. (iii) Find the trapezium rule estimate and show that it is the mean of the two bounds.

   [6 marks]

   (b)

   Find the percentage error in the 4-strip trapezium rule estimate. Use 8 strips of width 0.5 to find a better estimate, find its percentage error, and comment on the change in accuracy. Give percentages to 2 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions