Question

Find the area of the region between the curve y = x(x − 4)(x − 8) and the x-axis.

Answer 1

The total area of the regions between the curves is 128 square units

Calculating the total area of the regions between the curves

From the question, we have the following parameters that can be used in our computation:

y = x(x - 4)(x - 8)

On the x-axis, y = 0

So, the have the intersection to be

x(x - 4)(x - 8) = 0

Evaluate

x = 0, x = 4 and x = 8

Using the interval x = 0 and x = 4, the area of the regions between the curves is calculated as

`\text{Area} = 2 * \int\limits^4_0 {y} \, dx`

This gives

`\text{Area} = 2 * \int\limits^4_0 { x(x - 4)(x - 8)} \, dx`

Expand

`\text{Area} = 2 * \int\limits^4_0 { x^3 - 12x^2 + 32x} \, dx`

Integrate

`\text{Area} = 2 * [(x^4)/(4) - 4x^3 + 16x^2]|\limits^4_0`

Expand

`\text{Area} = 2 * ([(4^4)/(4) - 4 * 4^3 + 16 * 4^2] - [(0^4)/(4) - 4 * 0^3 + 16 * 0^2])`

Evaluate

`\text{Area} = 2 * (64 - 0)`

Area = 128

Hence, the total area of the regions between the curves is 128 square units