Double int x squared dx dy bounded by xy=16 and the line y=x, y=0 and x=8

We are tasked with evaluating the double integral \( \iint_D x^2 \, dx \, dy \), where the region \( D \) is bounded by the curves and lines:

- \( xy = 16 \) (hyperbola),
- \( y = x \) (diagonal line),
- \( y = 0 \) (x-axis), and
- \( x = 8 \) (vertical line).

### Step 1: Analyze the Region
The region \( D \) is defined by the hyperbola \( xy = 16 \), the line \( y = x \), and the boundaries \( y = 0 \) and \( x = 8 \). We need to describe the region in terms of appropriate limits of integration.

1. **For the hyperbola**: \( xy = 16 \) implies \( y = \frac{16}{x} \).
2. **For the line**: \( y = x \).
3. The region is bounded from \( y = 0 \) (the x-axis) up to the line \( y = x \), and we are only considering \( x \) from 4 to 8, since these are the bounds of the intersections of the curves within the region of interest.

### Step 2: Set up the Double Integral
We can express the region in terms of \( x \) limits and \( y \)-limits as follows:

- \( x \) ranges from 4 to 8.
- For a fixed \( x \), \( y \) ranges from \( 0 \) to \( \min(x, \frac{16}{x}) \), which in this case simplifies to \( y = \frac{16}{x} \) because \( \frac{16}{x} \leq x \) in the region between 4 and 8.

So, the integral becomes:
\[
\int_4^8 \int_0^{\frac{16}{x}} x^2 \, dy \, dx
\]

### Step 3: Solve the Integral

1. First, integrate with respect to \( y \):
\[
\int_0^{\frac{16}{x}} x^2 \, dy = x^2 \left[ y \right]_0^{\frac{16}{x}} = x^2 \cdot \frac{16}{x} = 16x
\]
   
2. Now, integrate with respect to \( x \):
\[
\int_4^8 16x \, dx = 16 \int_4^8 x \, dx = 16 \left[ \frac{x^2}{2} \right]_4^8 = 16 \cdot \frac{1}{2} \left( 8^2 - 4^2 \right)
\]
\[
= 8 \cdot (64 - 16) = 8 \cdot 48 = 384
\]

### Final Answer:
The value of the double integral is \( \boxed{384} \).

Would you like more details or have any questions about this?

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Here are 5 related questions:
1. What are the typical uses of double integrals in physics?
2. How would the answer change if the region were bounded by a different line, such as \( y = 2x \)?
3. How would you convert the problem to polar coordinates?
4. What is the geometric meaning of a double integral in terms of area or volume?
5. Can you generalize this approach for any curve \( xy = c \)?

**Tip:** Always sketch the region of integration first to ensure you understand the boundaries clearly.

Learn how to evaluate the double integral \( \iint_D x^2 \, dx \, dy \) where the region is bounded by the hyperbola xy = 16, the line y = x, and the boundaries y = 0 and x = 8. This step-by-step solution covers setting up the integral and solving it. Suitable for university-level students studying calculus.

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