Numerical integration with the trapezium ruleEdexcel A-Level Maths: Mind map

Exam questions on Numerical integration with the trapezium rule

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.
   Explain whether the estimate is an over-estimate or an under-estimate of $I$.2 marks
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 $x=0,\,0.25,\,0.5,\,0.75,\,1$.
   Find the trapezium rule estimate of $\int_0^1 g(x)\,\mathrm{d}x$.2 marks
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.
   Use the trapezium rule to estimate the distance travelled in the $8$ seconds.3 marks