L3 Integration
Question Bank
Use the values in the table below and the trapezium rule to find an approximation to $\displaystyle\int_0^{1.2} f(x)\,dx$.
$ \begin{array}{c|c} x & f(x) \\ \hline 0 & 3.0 \\ 0.2 & 3.4 \\ 0.4 & 3.6 \\ 0.6 & 3.5 \\ 0.8 & 3.1 \\ 1.0 & 2.6 \\ 1.2 & 1.9 \end{array} $
Use the values in the table below and Simpson's rule to find an approximation to $\displaystyle\int_1^3 f(x)\,dx$.
$ \begin{array}{c|c} x & f(x) \\ \hline 1 & 1.6 \\ 1.25 & 2.0 \\ 1.5 & 2.2 \\ 1.75 & 2.4 \\ 2 & 2.5 \\ 2.25 & 2.4 \\ 2.5 & 2.2 \\ 2.75 & 1.9 \\ 3 & 1.5 \end{array} $
Give your answer correct to 3 significant figures.
A cyclist starts from rest. Her speed, $v$ metres per second, is recorded every $5$ seconds for the first $30$ seconds:
$ \begin{array}{c|c} t\ (\text{s}) & v\ (\text{m/s}) \\ \hline 0 & 0 \\ 5 & 3.6 \\ 10 & 5.9 \\ 15 & 7.3 \\ 20 & 8.1 \\ 25 & 8.5 \\ 30 & 8.6 \end{array} $
Use the trapezium rule to estimate the distance she travels in the $30$ seconds.
The diagram shows the cross-section of a river that is $28$ m wide. The depth of the water was measured every $4$ m across the river.
Use the trapezium rule to find an approximation for the area of the cross-section.
The diagram shows a yacht's sail. Its straight edge runs $7.2$ m up the mast, and it comes to a point at the top of the mast. Its width, measured at right angles to the mast, was recorded every $1.2$ m up from the boom.
Use the trapezium rule to find an approximation for the area of the sail.
A small field lies between a straight fence $40$ m long and a stream. At the left-hand end there is a short straight fence at right angles to the long one; at the right-hand end the stream meets the long fence. The width of the field was measured every $5$ m along the long fence.
Use Simpson's rule to find an approximation for the area of the field. Give your answer correct to 3 significant figures.
Sione is building a kayak. The diagram shows its deck, seen from above. The deck is $4$ m long and pointed at both ends, and its width was measured every $0.5$ m along its length.
Use Simpson's rule to find an approximation for the area of the deck, correct to 3 significant figures.
A stopbank is a bank of earth built along a river to hold back floodwater. The diagram shows the cross-section of a stopbank, which is $15$ m wide at its base. Its height was measured every $2.5$ m across the base.
(a) Use Simpson's rule to find an approximation for the area of the cross-section, correct to 3 significant figures.
(b) A $40$ m length of the stopbank has this cross-section all the way along. Estimate the volume of earth in it, correct to 3 significant figures.
The entrance to a road tunnel is an arch over a flat road. The arch is $8$ m wide at road level, and its height above the road was measured every $2$ m across, as shown.
(a) Use the trapezium rule to estimate the area of the tunnel entrance.
(b) An engineer models the arch by
$ y = 3x - \frac{3x^2}{8}, \quad 0 \le x \le 8, $
where $x$ m is the distance across from the left-hand side and $y$ m is the height above the road. Use integration to find the exact area under the model, and compare it with your estimate from (a).
You must use calculus and show the results of any integration needed to solve the problem.