4. $[-12$ Points $]$
DETAILS SCALC9 5.1.010.
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Set up an integral representing the area $A$ of the region enclosed by the given curves.
\[
x=y^{4}, x=2-y^{2}
\]
\[
A=\int_{-1}(\square) d y
\]
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Answer
Final Answer: The area of the region enclosed by the curves $x=y^{4}$ and $x=2-y^{2}$ is \(\boxed{\frac{44}{15}}\).
Step 1 ：First, we need to find the intersection points of the two curves $x=y^{4}$ and $x=2-y^{2}$. The intersection points are -1 and 1. The other two points are imaginary and are not considered in this context.
Step 2 ：Next, we set up the integral from -1 to 1 (the intersection points). The integrand will be the absolute difference between the two functions. We need to determine which function is greater in the interval from -1 to 1.
Step 3 ：Finally, we calculate the integral to find the area of the region enclosed by the curves. The area is \(\frac{44}{15}\).
Step 4 ：Final Answer: The area of the region enclosed by the curves $x=y^{4}$ and $x=2-y^{2}$ is \(\boxed{\frac{44}{15}}\).
