Trapezium rule

The curve with equation y = is shown in the diagram.

The region shaded is bounded by the curve, the x-axis, and the lines x = 4 and x = 8

The trapezium rule with six ordinates (5 strips) is to be used to find an approximate value for the area of the shaded region.

Some of the values required to obtain this approximation are shown in the table below.

Find the y-value that is missing from the table.

Use the trapezium rule with six ordinates (5 strips) to find an approximate value for the area of the shaded region.

Give your answer to five significant figures.

A student finds an improved approximation for the area of the shaded region by using the trapezium rule with 11 ordinates.

The student correctly obtains 29.759 as their improved approximation.

The student claims that the exact area must be greater than 29.759

Without further calculation, explain whether or not the student is correct.

Figure 2 below shows a 1.5 metre length of pipe.

The symmetrical cross-section of the pipe is shown below, in Figure 3, where x and y are measured in centimetres.

Use the trapezium rule, with the values shown in the table below, to find the best estimate for the volume of the pipe.

Use a suitable substitution to show that

can be written as

where a is a constant to be found.

Hence, or otherwise, show that

A graph has the equation

A student uses four rectangles to approximate the area under the graph between the lines x = 0 and x = 4

The rectangles are all the same width.

All the rectangles are drawn under the curve as shown in the diagram below.

The total area of the four rectangles is A

The student decides to improve their approximation by increasing the number of rectangles used.

Explain why the value of the student’s improved approximation will be greater than A, but less than

The trapezium rule with 6 ordinates (5 strips) is to be used to find an approximation for the shaded area.

The values required to obtain this approximation are shown in the table below.

Use the trapezium rule with 6 ordinates (5 strips) to find an approximate value for the shaded area.

Give your answer to four decimal places.

Using your answer to part (a) deduce an estimate for dx

The trapezium rule is used to estimate the area of the shaded region in each of the graphs below. Identify the graph for which the trapezium rule produces an overestimate.

Tick () one box.

The region bounded by the curve

and the x-axis is shaded in the diagram below.

Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region.

Give your answer correct to three significant figures.

Show that the exact area is given by

Fully justify your answer.

t seconds as shown by the following graph.

Noosha says that, in the period 12 36, the scooter travels approximately 130 metres.

Determine if Noosha is correct, showing clearly any calculations you have used.

by the graph below.

Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period 20 t 100

Over the same time period, the curve can be very closely modelled by a particular quadratic.

Explain how you could find an alternative estimate using this quadratic.

Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into n trapezia.

When n 4

State the number of ordinates that Caroline uses.

Calculate the area that Caroline should obtain using this method.

Give your answer correct to two decimal places.

Show that the exact area of A is

Fully justify your answer.

Explain what would happen to Caroline’s answer to part (a)(ii) as n