y = 16/x2 - x/2 + 1
The finite region R, bounded by the lines x = 1, the x-axis and the curve, is shown shaded in Figure 1. The curve crosses the x-axis at the point4, 0).
(a) Complete the table with the values of y corresponding to x = 2 and 2.5
(b) Use the trapezium rule with all the values in the completed table to find an approximate value for the area of R, giving your answer to 2 decimal places.
(c) Use integration to find the exact value for the area of R.
6 (a)(b) Trapezium rule.
Figure 2 shows ABC, a sector of a circle of radius 6 cm with centre A. Given that the size of angle BAC is 0.95 radians, find
(a) the length of the arc BC,
(b) the area of the sector ABC.
The point D lies on the line AC and is such that AD BD = . The region R, shown shaded in Figure 2, is bounded by the lines CD, DB and the arc BC.
(c) Show that the length of AD is 5.16 cm to 3 significant figures.
(d) the perimeter of R,(e) the area of R, giving your answer to 2 significant figures.
Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius x metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to x metres and width equal to y metres.
Given that the area of the flowerbed is 4 m2
(a) show that
y = (16 - πx2)/8x
(b) Hence show that the perimeter P metres of the flowerbed is given by the equation
P = 8/x + 2x
(c) Use calculus to find the minimum value of P.
(i) Find the solutions of the equation sin(3x - 15°) = 1/2, for which 0° ≤ x ≤ 180°
Figure 4 shows part of the curve with equation
y = sin(ax - b), where a > 0, 0 < b < π
The curve cuts the x-axis at the points P, Q and R as shown.
Given that the coordinates of P, Q and R are (π/10, 0), (3π/5, 0), 11π/10, 0) respectively, find the values of a and b.9 (i)
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