Find the signed area between the 𝑥-axis and the graph of 𝑦=𝑥2−4 over the interval [2,4]

To find the signed area between the x-axis and the graph of y=x^2-4 over the interval [2,4], we need to integrate the function from 2 to 4.

The integral of the function y=x^2-4 from 2 to 4 is:

∫ from 2 to 4 (x^2 - 4) dx

This is a simple power rule problem in calculus. The power rule states that the integral of x^n dx is (1/(n+1))x^(n+1).

So, the antiderivative of x^2 is (1/3)x^3 and the antiderivative of 4 is 4x.

Therefore, the antiderivative of (x^2 - 4) is (1/3)x^3 - 4x.

We evaluate this from 2 to 4 to get:

[(1/3)4^3 - 44] - [(1/3)2^3 - 42]

Simplify that to get:

[(64/3 - 16] - [(8/3 - 8)]

Which simplifies to:

[64/3 - 48/3] - [8/3 - 24/3]

Further simplification gives:

[16/3] - [-16/3]

So, the signed area between the x-axis and the graph of y=x^2-4 over the interval [2,4] is 32/3 or approximately 10.67 square units.