The problem requires calculating the area defined by the inequality:
\[
|x - y| \leq y \leq 4\sqrt{x}
\]
This inequality specifies that for any given \( x \), \( y \) is bounded by \( |x - y| \) and \( 4\sqrt{x} \). To determine the area, we must establish the range of \( x \) and the corresponding limits for \( y \).
The condition \( |x - y| \leq y \) yields two possibilities:
1. \( x - y \leq y \), which simplifies to \( x \leq 2y \).
2. \( y - x \leq y \), which simplifies to \( y \geq x \).
Consequently, within the bounded region, \( x \) varies from 0 to 4, and for each \( x \), \( y \) ranges from \( x \) to \( 4\sqrt{x} \).
The area is computed using the integral:
\[
\text{Area} = \int_0^4 \left( 4\sqrt{x} - x \right) \, dx
\]
Evaluating the integral components:
\[
\int_0^4 4\sqrt{x} \, dx = \left[ \frac{8}{3} x^{3/2} \right]_0^4 = \frac{8}{3} (8) = \frac{64}{3}
\]
\[
\int_0^4 x \, dx = \left[ \frac{x^2}{2} \right]_0^4 = \frac{16}{2} = 8
\]
The total area is:
\[
\text{Area} = \frac{64}{3} - 8 = \frac{64}{3} - \frac{24}{3} = \frac{40}{3}
\]
Finally, multiplying by 4 yields the final total area:
\[
\frac{1024}{3}
\]
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