Trapezium rule

Use the trapezium rule, with intervals of 0.5, to determine an approximation to the area

of the shaded region.

Use rectangles with the same intervals as in part (i) to determine lower and upper bounds

for the exact area of the shaded region.

Write an expression for the exact area of the shaded region in terms of a limit, involving x

and intervals of width δ x.

Give your answer correct to 3 significant figures.

State whether the value from part (a) is an under-estimate or an over-estimate, giving a

reason for your answer.

Explain how the trapezium rule could be used to obtain a more accurate estimate.

Alex is investigating the area, A, under the graph of y = x² between x = 1 and x = 1.5. They draw the graph, together with rectangles of width δ x = 0.1, and varying heights y.

Use the rectangles in the diagram to show that lower and upper bounds for the area A are 0.73

and 0.855 respectively.

Write down an expression, in terms of y and δ x, for the exact value of the area A.

The diagram shows the part of the curve y = 3x sin 2x for which 0 ⩽ x ⩽ .

The maximum point on the curve is denoted by P.

Show that the x-coordinate of P satisfies the equation tan 2 x + 2 x = 0.

Use the Newton-Raphson method, with a suitable initial value, to find the x-coordinate of P,

giving your answer correct to 4 decimal places. Show the result of each iteration.

The trapezium rule, with four strips of equal width, is used to find an approximation to

Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under-or over-estimate for the area of the

region in this case.

Use the trapezium rule, with two strips of equal width, to show that

Use the substitution x = u² to find the exact value of

Using your answers to parts (i) and (ii), show that

where k is a rational number to be determined.