Key Concepts and Formulas

• Area Between Curves: The area between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$, where $f(x) \geq g(x)$ on $[a, b]$, is given by $\int_a^b [f(x) - g(x)] dx$.
• Absolute Value Inequality: The inequality $|x| \leq a$ is equivalent to $-a \leq x \leq a$.
• Power Rule of Integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.

Step-by-Step Solution

Step 1: Deconstruct the Inequalities

We are given the region defined by $\{(x, y):|x-y| \leq y \leq 4 \sqrt{x}\}$. We need to analyze the inequalities to determine the bounding curves.

• The inequality $y \leq 4\sqrt{x}$ implies that $x \geq 0$ and $y \geq 0$ since the square root function returns a non-negative value.

• The inequality $|x-y| \leq y$ can be rewritten as $-y \leq x-y \leq y$. This breaks down into two inequalities:

  • $-y \leq x-y$. Adding $y$ to both sides yields $0 \leq x$, or $x \geq 0$.
  • $x-y \leq y$. Adding $y$ to both sides yields $x \leq 2y$, which implies $y \geq \frac{x}{2}$.

Therefore, the region is bounded by $y \leq 4\sqrt{x}$ (upper curve) and $y \geq \frac{x}{2}$ (lower curve), with $x \geq 0$.

Step 2: Find the Intersection Points

To find the limits of integration, we need to find the intersection points of $y = 4\sqrt{x}$ and $y = \frac{x}{2}$. Setting these equal to each other gives: $4\sqrt{x} = \frac{x}{2}$ Squaring both sides: $(4\sqrt{x})^2 = \left(\frac{x}{2}\right)^2$ $16x = \frac{x^2}{4}$ $64x = x^2$ $x^2 - 64x = 0$ $x(x - 64) = 0$ This gives $x = 0$ or $x = 64$.

When $x=0$, $y = \frac{0}{2} = 0$. When $x=64$, $y = \frac{64}{2} = 32$.

The intersection points are $(0, 0)$ and $(64, 32)$. So, the limits of integration are $x=0$ and $x=64$.

Step 3: Set Up the Definite Integral

The area of the region is given by the integral: $\text{Area} = \int_0^{64} \left(4\sqrt{x} - \frac{x}{2}\right) dx$ We can rewrite this as: $\text{Area} = \int_0^{64} \left(4x^{1/2} - \frac{1}{2}x\right) dx$

Step 4: Evaluate the Definite Integral

Now we integrate: $\text{Area} = \left[4 \cdot \frac{x^{3/2}}{3/2} - \frac{1}{2} \cdot \frac{x^2}{2}\right]_0^{64}$ $\text{Area} = \left[\frac{8}{3}x^{3/2} - \frac{1}{4}x^2\right]_0^{64}$ Evaluating at the limits: $\text{Area} = \left(\frac{8}{3}(64)^{3/2} - \frac{1}{4}(64)^2\right) - \left(\frac{8}{3}(0)^{3/2} - \frac{1}{4}(0)^2\right)$ $\text{Area} = \frac{8}{3}(8^3) - \frac{1}{4}(4096)$ $\text{Area} = \frac{8}{3}(512) - 1024$ $\text{Area} = \frac{4096}{3} - 1024$ $\text{Area} = \frac{4096}{3} - \frac{3072}{3}$ $\text{Area} = \frac{1024}{3}$

Common Mistakes & Tips

• Remember to correctly identify the upper and lower curves. A sketch can be very helpful.
• When squaring equations to eliminate square roots, check for extraneous solutions.
• Be careful with the arithmetic, especially when dealing with fractions.

Summary

We deconstructed the given inequalities to find the bounding curves, determined the intersection points to establish the limits of integration, set up the definite integral, and evaluated it to find the area of the region. The area of the region is $\frac{1024}{3}$ square units.

Final Answer

The final answer is $\boxed{\frac{1024}{3}}$, which corresponds to option (D).