8.3 Areas Under the X-axis

Sketch the following and find the total area of the finite region or regions bounded by the curves and the x-axis:

a) \(y = x(x + 2)\)

b) \(y = (x + 1)(x - 4)\)

c) \(y = (x + 3)x(x - 3)\)

d) \(y = x^2(x - 2)\)

e) \(y = x(x - 2)(x - 5)\)

The graph shows a sketch of part of the curve C with equation \(y = x(x + 3)(2 - x)\).

The curve C crosses the x-axis at the origin O and at points A and B.

a) Find the coordinates of point A and point B. (2 marks)

b) Find the area of the finite region bounded by C and the x-axis. (4 marks)

The shaded area under the graph of the function \(f(x) = 3x^2 - 2x + 2\), bounded by the curve, the x-axis and the lines x = 0 and x = k, is 8. Work out the value of k.

The finite region R is bounded by the x-axis and the curve with equation \(y = -x^2 + 2x + 3, x \geq 0\).

The curve meets the x-axis at points A and B.

a) Find the coordinates of point A and point B. (2 marks)

b) Find the area of the region R. (4 marks)

The graph shows part of the curve C with equation \(y = x^2(2 - x)\).

The region R, shown shaded, is bounded by C and the x-axis. Use calculus to find the exact area of R. (5 marks)