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This problem is asking which of the given integrals represents a volume. Let's break down the given integral in part (a).

The integral is:

\[
J_6 = \iint_D (xy^2 + 1) \, dA
\]
where \( D = \{ (x, y) \mid -4 \leq x \leq 4, \, x^2 \leq y \leq 4 - x^2 \} \).

### Interpretation:
1. **Region \( D \):** The region \( D \) is bounded by \( -4 \leq x \leq 4 \), and for each \( x \), \( y \) varies between \( x^2 \) and \( 4 - x^2 \).
   - The upper bound for \( y \) is \( 4 - x^2 \), which describes a downward-facing parabola. 
   - The lower bound is \( x^2 \), which describes an upward-facing parabola. The region between these parabolas forms a closed area, describing the 2D domain in which the integral is computed.

2. **Integrand \( xy^2 + 1 \):** This function defines the "height" of the surface above the domain \( D \). The constant term \( 1 \) guarantees that the integrand is positive over the region, implying it measures something akin to a volume.

### Volume Check:
An integral represents a volume if it integrates a positive function over a two-dimensional region, where the integrand defines the height above that region. In this case, the function \( xy^2 + 1 \) is the "height" over the region \( D \), and since the constant term \( +1 \) ensures non-negativity, this does represent a volume.

Thus, **integral (a) represents a volume**.

Would you like a more detailed explanation or have any specific questions? Here are some related questions that could deepen understanding:

1. How do the bounds of the region \( D \) affect the shape of the region?
2. What is the significance of the integrand having a constant \( +1 \) term?
3. How would the interpretation change if the integrand were purely a function of \( x \) and \( y \), without the \( +1 \)?
4. Could this integral represent something other than volume under different conditions?
5. How does this integral differ from a triple integral when calculating volume?

**Tip:** Always analyze the integrand and the domain carefully to understand whether the integral is computing area, volume, or another quantity.

Lesquelles des intégrales suivantes représentent un volume? (a) J6 = ∫∫D (xy² + 1) dA, où D = { (x, y) | -4 ≤ x ≤ 4, x² ≤ y ≤ 4 - x² }.

Learn how to determine whether the given double integral represents a volume, with a focus on analyzing the bounds and the integrand function. This example features a surface defined by xy² + 1 over a domain bounded by parabolas. Suitable for advanced calculus students.

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