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A triple integral involves integrating a function in three-dimensional space, which allows us to compute the volume or the integral of a function over a solid region. In problems involving triple integrals, our goal is to integrate a function over a three-dimensional region, denoted as \( S \). This requires iterating multiple integrals one after another.
To evaluate a triple integral like \( \iiint_{S} f(x, y, z) \ dV \), we integrate in three stages. The process involves choosing a specific order for variables \( x \), \( y \), and \( z \). The choice of order depends on the complexity of the limits that define the region. Here, the most straightforward order, as shown in the example, is \( z \), \( x \), and then \( y \).
When executing a triple integral:

• Choose integration limits for each variable.
• Evaluate the innermost integral first, followed by the next layer, until all integrations are completed.
• Ensure that your chosen order simplifies the boundaries and simplifies overall calculations.

By setting up and evaluating triple integrals, we can determine functions' "accumulated鈥� values over three-dimensional spaces.

Sketching the solid defined by certain inequalities is crucial for understanding the bounds of integration. In the given problem, the solid \( S \) is defined by three constraints: \( 0 \leq x \leq \sqrt{y} \), \( 0 \leq y \leq 4 \), and \( 0 \leq z \leq \frac{3}{2} x \).
Start by sketching the region on the \( xy \)-plane. This involves drawing the area enclosed by the boundaries. Here, the inequality \( 0 \leq x \leq \sqrt{y} \) suggests looking at the square root function in reverse, forming a right parabolic region under the curve \( y = x^2 \). For \( x \) from 0 to 2 when \( y \) reaches 4, this parabolic region captures the horizontal limits of the solid.
Next, add the vertical \( z \)-dimension. With \( 0 \leq z \leq \frac{3}{2} x \), visualize this as triangular cross-sections expanding upwards. Each cross-section's height is proportional to \( x \), creating a sloped surface that extends over the region defined in the \( xy \)-plane. This visualization helps confirm the integration boundaries.

• Draw the base in the \( xy \)-plane.
• Add the vertical dimension to showcase the solid volume your integral covers.
• Use visualization to simplify recognizing limits for integration.

Understanding and setting integration bounds correctly ensures that you evaluate the integral over the entire desired region. In the exercise, our solid is bounded by:

• \( 0 \leq x \leq \sqrt{y} \)
• \( 0 \leq y \leq 4 \)
• \( 0 \leq z \leq \frac{3}{2}x \)

Start by identifying the integration limits for each variable, beginning with the innermost integral and moving outward. In the provided solution, \( z \) is integrated first from \( 0 \) to \( \frac{3}{2}x \). This corresponds to the slanted upper boundary of the solid.
Next, \( x \) is integrated from \( 0 \) to \( \sqrt{y} \), covering the horizontal range of the parabolic base in the \( xy \)-plane. The parabolic shape arises from the square root's influence on \( x \), which restricts x's maximum value based on \( y \).
Finally, \( y \) is integrated from 0 to 4, exploring the entire vertical range of the base plane. Determine the appropriate limits through both visualization and the constraints themselves.

• Understand every variable's range through the solid's physical constraints.
• Verify bounds through sketching and matching them with inequality constraints.
• Adjust integration limits to match the computed or visualized boundaries accurately.

Finding these boundaries helps set up the iterated integral correctly and solve the problem precisely.