https://file.aiquickdraw.com/mathbot/file/1729315295972mxsgqqth.jpg

Let's analyze the problem.

We are asked to find the volume \( V \) of the solid \( Q \), which is bounded by the following surfaces:
- The cylinder \( y = x^2 \),
- The planes \( y + z = 1 \) and \( z = 0 \) (i.e., the base of the solid is on the \( xy \)-plane).

### Step-by-Step Approach

#### 1. Understanding the boundaries:
- The **cylinder** \( y = x^2 \) describes the lateral boundary.
- The **plane** \( y + z = 1 \) can be rewritten as \( z = 1 - y \), which is the top boundary of the solid.
- The plane \( z = 0 \) is the base of the solid.

Thus, the solid is bounded from below by \( z = 0 \) and from above by \( z = 1 - y \), where \( y \) itself is constrained by the cylinder \( y = x^2 \).

#### 2. Setting up the double integral:
To find the volume, we can integrate the height of the solid \( z = 1 - y \) over the region bounded by the projection of the solid on the \( xy \)-plane.

Since \( y = x^2 \) is the boundary for \( y \), we can express the region as:
- \( x \) ranges from \( -1 \) to \( 1 \) (based on the limits given by the cylinder).
- For each \( x \), \( y \) ranges from \( x^2 \) to \( 1 \) (from the cylinder up to the plane).

Therefore, the volume integral should be:

\[
V = 2 \int_0^1 \int_{x^2}^{1} (1 - y) \, dy \, dx
\]

This matches the third option:
\[
V = 2 \int_0^1 \int_{\sqrt{y}}^{1} (1 - x^2) \, dx \, dz
\]

Thus, the correct answer is the third option (highlighted in the image).

Would you like a detailed breakdown of the solution? Let me know if you have any questions!

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Here are 5 related questions for further exploration:
1. How do you handle triple integrals when finding volumes bounded by more complex surfaces?
2. What is the general strategy for converting to cylindrical or spherical coordinates in volume integrals?
3. How can this problem be solved using symmetry properties?
4. How do we set up the boundaries for more irregular shapes bounded by multiple surfaces?
5. What would change if the boundaries were expressed in polar coordinates?

**Tip**: Always visualize the region of integration when setting up a double or triple integral. Drawing the projection on the \( xy \)-plane helps in identifying the correct bounds.

Se quiere encontrar el volumen del sólido Q que está acotado por el cilindro y = x^2 y los planos y + z = 1 y z = 0.

Learn how to calculate the volume of a solid bounded by the cylinder y = x^2 and the planes y + z = 1 and z = 0 using double integrals. This problem involves setting up a double integral with appropriate limits and evaluating the result.

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