Question

Find the area between y=-x^{4}+4 x²+3, y=x-1, and ( -1.8 ≤ x ≤1.8 ). Round your limits of integration and answer to 2 decimal places. The area between the curves is square units.

Answer 1

To calculate the area between the curves y=-x^4+4x^2+3 and y=x-1 over the range (-1.8 ≤ x ≤ 1.8) in square units, we approximate the limits as -2 and 2 and integrate the difference of the functions over that range.

Step-by-step explanation:

To find the area between the curves y=-x^{4}+4x^{2}+3, y=x-1, and the range (-1.8 ≤ x ≤ 1.8), we need to set up the integral of the difference between these two functions over the given range. However, since the limits of integration must be integers, we round them off to -2 and 2. We then calculate the integral of the top function minus the bottom function.

The integral to find the area A would be:

A = ∫_{-2}^{2} [(x-1)-(-x^{4}+4x^{2}+3)] dx

Simplifying the integrand gives us:

A = ∫_{-2}^{2} (x^{4}-4x^{2}+x-4) dx

Applying the Fundamental Theorem of Calculus, we find the antiderivatives and evaluate at the limits of integration to find the area.

It's critical to recognize that the specific functions and limits provide the context needed to evaluate this as an area problem, similar to how the area under a curve can represent different physical concepts like work or probability, depending on the variables involved and the physical interpretation.