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The problem in the image involves a solid described by the equation \( x^2 + 4y^2 + z^2 = 36 \), and it is divided into three parts:

### (a) Sketch the solid \( U \)
The equation \( x^2 + 4y^2 + z^2 = 36 \) represents an ellipsoid. Since the coefficients of \( x^2 \), \( 4y^2 \), and \( z^2 \) are different, the ellipsoid is stretched along the \( y \)-axis. Specifically:
- In the \( x \)-direction: \( -6 \leq x \leq 6 \)
- In the \( y \)-direction: \( -3 \leq y \leq 3 \) (since \( 4y^2 \) gives half the radius in the \( y \)-axis)
- In the \( z \)-direction: \( -6 \leq z \leq 6 \)

The condition \( -2 \leq y \leq 2 \) further restricts the solid in the \( y \)-direction. So the solid is a truncated ellipsoid, symmetrical in \( x \) and \( z \), and bounded by \( y = \pm 2 \).

A sketch would show an ellipsoid that is longer along the \( x \)- and \( z \)-axes but is "cut off" at \( y = -2 \) and \( y = 2 \).

### (b) Is \( \iiint_U (z - 3) \, dV \) positive, negative, or zero?
The expression \( z - 3 \) shifts the \( z \)-coordinate by 3 units. To determine whether the integral is positive, negative, or zero, we need to consider the symmetry of the ellipsoid and the behavior of \( z - 3 \):

- The solid is symmetric around \( z = 0 \), but \( z - 3 \) is biased towards negative values for most of the volume because \( z \leq 3 \) for a large portion of the region.
- Therefore, the integral is likely to be negative, as the volume where \( z - 3 \) is negative will dominate.

### (c) Evaluate \( \iiint_U (z - 3) \, dV \)
To evaluate this integral, you need to set up the iterated integral based on the bounds of the ellipsoid. Converting to cylindrical coordinates might simplify the integration process, given the symmetry. The steps to set it up in cylindrical coordinates would involve transforming \( x^2 + 4y^2 + z^2 = 36 \) and solving accordingly.

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Do you want a more detailed explanation for any specific part or help setting up the integrals explicitly?

### Related Questions:
1. What does the ellipsoid equation represent geometrically?
2. How does symmetry help determine whether an integral will be positive or negative?
3. Can cylindrical coordinates always simplify integrals with elliptical boundaries?
4. What conditions affect whether the integral \( \iiint_U (z - 3) dV \) is zero?
5. What would happen if the limits on \( y \) were changed from \( -2 \leq y \leq 2 \)?

**Tip:** When evaluating triple integrals, always consider using symmetries in the problem to simplify the bounds and integrand.

Let U be the solid consisting of all points inside x^2 + 4y^2 + z^2 = 36 with -2 <= y <= 2 and z >= 0. (a) Sketch U. (b) Without evaluating, is ∫∫∫U (z - 3) dV positive, negative, or 0? Explain how you can be sure without evaluating. (c) Evaluate ∫∫∫U (z - 3) dV. Please write out your iterated integral clearly; you’re welcome to use WolframAlpha to evaluate it.

Explore the step-by-step guide to sketching and evaluating a volume integral over the ellipsoid defined by x^2 + 4y^2 + z^2 = 36, confined within specific bounds. This in-depth explanation covers the necessary mathematical concepts, including ellipsoids and iterated integrals, suitable for university-level calculus courses.

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