(ii) Find the exact area between the curve
y = 2cos 2x + 1 and the line y = 1
from x = 0 to x = π/3.

(i) State the range of f(x);

(ii) Sketch f(x) for 0 ≤ x ≤ 2π;

(iii) Calculate the exact area of the region bounded by the curve
y = 2cos x + 1, the y axis and the line y = 1.

(ii) Find the area enclosed between the curve and the x axis between x = 0 and
x = π.

Calculate the exact area between the curve and the x axis from x = -π/2 to π/2.

(i) Solve f(x) = 0.

(ii) Find any stationary points on f(x) and determine their nature.

(iii) Sketch the graph of y = f(x).

(iv) Find the area bounded by the curve and the x-axis from

8. The breathing cycle for an adult at rest takes 5 seconds on average.

For the first 2.5 seconds (0 < t ≤ 2.5), breath is taken into the lungs. For the remaining 2.5 seconds (2.5 < t ≤ 5), breath is exhaled.

The rate (R litres/second) at which breath is being inhaled into or exhaled from the lungs at time t through the 5 second cycle can be modelled by the equation

How many litres of air is inhaled into the lungs during one breathing cycle?

(ii) Find the area bounded by these curves, the y axis and the line x = π/4.

(ii) Find the area between the two curves within the domain [0, π/3].

(i) Show that the point of intersection has coordinates (1.5, 1).

(ii) Find the exact area described the the y axis and the area between the two curves up their point of intersection.

The diagram shows the region bounded by the curves y = sec² x, y = 2 cot x and the coordinate axes.

(i) Show by substitution that the point lies on both y = sec² x
and y = 2 cot x.

(ii) Differentiate ln (sin x).

(iii) Hence or otherwise, find the exact area of the shaded region.

(ii) Find the area bounded between the two curves in the given domain.

14. (i) Sketch the curves y = sin 2x and y = cos x in the domain [-π/2, π/2].

(ii) fine the total area enclosed by these two curves in the given domain.

The diagram above shows the graphs of y = sin x and y = √3 cos x (0 ≤ x ≤ 2π).

The graphs intersect at points A and B.

(i) Find the x coordinates of points A and B.

(ii) Find the area enclosed by the two graphs.

(ii) Find the area between these two curves between the two points of intersection
(correct to 3 decimal places) .