We are being asked to find the area under the given function for the specified intervals. Note that since we are computing the area between the function and the x axis, all we need to do is computer the definite integral as we normally would.

Step 1:

The definite integral is as follows:

[latex]\large{\int_{2}^{4}(x^2-2x)dx = (\frac{1}{3}x^3 – x^2)|_{2}^{4}}[/latex]

Step 2:

Now we can apply the fundamental theorem of calculus to evaluate. This states that:

[latex]\large{\int_{a}^{b}f(x) dx = F(b) – F(a)}[/latex]

Applying the fundamental theorem of calculus, we have:

[latex]\large{\int_{2}^{4}(x^2-2x) dx}[/latex]

[latex]\large{ = [\frac{1}{3}(4)^3 – (4)^2] – [\frac{1}{3}(2)^3 – (2)^2]}[/latex]

[latex]\large{ \approx 5.33 – (- 1.33)}[/latex]

[latex]\large{\approx 6.67}[/latex]

Thus, the area between the function [latex]y = x^2-2x[/latex] and the x axis is approximately 6.67 units.

Solution Completed!