Numerical integration with the trapezium ruleEdexcel A-Level Maths: Subtopic test

1. 1
   The integral $I=\int_0^2\sqrt{x^2+1}\,\mathrm{d}x$ is estimated using the trapezium rule with $4$ strips of equal width.

   (a)

   Which expression gives the trapezium rule estimate of $I$?

   [1 mark]

   • A$\frac{0.5}{2}\left[1+\sqrt5+2\left(\sqrt{1.25}+\sqrt2+\sqrt{3.25}\right)\right]$
   • B$0.5\left[1+\sqrt5+2\left(\sqrt{1.25}+\sqrt2+\sqrt{3.25}\right)\right]$
   • C$\frac{0.5}{2}\left[1+\sqrt5+\sqrt{1.25}+\sqrt2+\sqrt{3.25}\right]$
   • D$\frac{0.5}{2}\left[2(1+\sqrt5)+\sqrt{1.25}+\sqrt2+\sqrt{3.25}\right]$

   (b)

   Find the value of the estimate, correct to 3 decimal places.

   [1 mark]

   • A$2.958$
   • B$5.953$
   • C$2.977$
   • D$1.893$

   (c)

   Explain whether the estimate is an over-estimate or an under-estimate of $I$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=2x+1$ and $g(x)=(2x+1)^2$ for $0\le x\le1$. The trapezium rule is used with $4$ strips, at $x=0,\,0.25,\,0.5,\,0.75,\,1$.

   (a)

   Find the trapezium rule estimate of $\int_0^1 f(x)\,\mathrm{d}x$.

   [1 mark]

   • A$1.6$
   • B$2$
   • C$16$
   • D$2.375$

   (b)

   Why is the estimate of $\int_0^1 f(x)\,\mathrm{d}x$ equal to the exact value?

   [1 mark]

   • AFour strips give an exact answer for any function.
   • B$f$ is an increasing function.
   • CThe graph of $f$ is concave.
   • DThe graph of $f$ is a straight line, so the trapezia match the area exactly.

   (c)

   Find the trapezium rule estimate of $\int_0^1 g(x)\,\mathrm{d}x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The speed of a cyclist, in m s$^{-1}$, is measured every $2$ seconds for $8$ seconds, giving $0,\ 4.5,\ 7.5,\ 9.0,\ 9.5$. The cyclist's speed is increasing throughout, at a decreasing rate.

   (a)

   Use the trapezium rule to estimate the distance travelled in the $8$ seconds.

   [3 marks]

   (b)

   Find lower and upper limits between which the distance must lie using rectangles, and use the shape of the speed-time data to say whether $51.5$ m is an over-estimate or an under-estimate. What can you say about the true distance?

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The integral $I=\int_1^3\frac1x\,\mathrm{d}x=\ln3$ is estimated using the trapezium rule.

   (a)

   Use the trapezium rule with $4$ strips to estimate $I$ to 4 decimal places. Explain why your estimate is an over-estimate, and compare it with the exact value.

   [6 marks]

   (b)

   The estimate is repeated with $8$ strips. Find it to 4 decimal places and evaluate how the accuracy changes when the number of strips is doubled.

   [6 marks]

   Total for question 4: 12 marks

End of questions