8.2 Areas Under Curves

Find the area between the curve with equation \(y = f(x)\), the x-axis and the lines \(x = a\) and \(x = b\) in each of the following cases:

a) \(f(x) = -3x^2 + 17x - 10\); \(a = 1, b = 3\)

b) \(f(x) = 2x^3 + 7x^2 - 4x\); \(a = -3, b = -1\)

c) \(f(x) = -x^4 + 7x^3 - 11x^2 + 5x\); \(a = 0, b = 4\)

d) \(f(x) = \frac{8}{x^2}\); \(a = -4, b = -1\)

The sketch shows part of the curve with equation \(y = x(x^2 - 4)\). Find the area of the shaded region.

The diagram shows a sketch of the curve with equation \(y = 3x + \frac{6}{x^2} - 5, x > 0\).

The region R is bounded by the curve, the x-axis and the lines \(x = 1\) and \(x = 3\). Find the area of R.

Find the area of the finite region between the curve with equation \(y = (3 - x)(1 + x)\) and the x-axis.

Find the area of the finite region between the curve with equation \(y = x(x - 4)^2\) and the x-axis.

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 \(k\).