Answer
The area between the curves can be found by a single integral as on the interval [-1,1], 1-$x^{2}$ is always $\geq$ $x^{4}-2x^{2}+1$
The integral for this area is:
\[
A = \int_{-1}^{1}[(1-x^{2})-(x^{4}-2x^{2}+1)]dx
\]
Work Step by Step
To find the integral for this area we need to set the two equations to each other and solve for the x-intercepts which will then be the lower and upper limits of our integral.
\[
1- x^2 = x^4 - 2x + 1
\]
\[
- x^2 = x^4 - 2x^2
\]
\[
0 = x^4 - x^2
\]
\[
0 = x^2 (x^2 - 1)
\]
\[
x^2 (x+1)(x-1) = 0
\]
\[
x = 0, -1, 1
\]
Using desmos.com we can graph the two functions to determine which function is on the top.
After looking at the graph we can determine the integral by subtracting the bottom function from the top function:
\[
A = \int_{-1}^{1}[(1-x^{2})-(x^{4}-2x^{2}+1)]dx
\]
