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The region of integration is a fundamental aspect of setting up a triple integral. In this case, the region \( E \) is defined by the set \( \{(x, y, z) | 0 \leq x \leq 1, -x^2 \leq y \leq x^2, 0 \leq z \leq 1\} \). Imagine a 3D space where every point \((x, y, z)\) must adhere to these limits to belong in the region \( E \).

• **X-Direction:** Here, \( x \) varies from 0 to 1. This indicates a line segment along the x-axis, restricting the entire region horizontally.
• **Y-Direction:** For each fixed \( x \), \( y \) stretches from \(-x^2\) to \(x^2\). Notice that this creates a varying range depending on \( x \), typically forming a parabola when visualized in the xy-plane.
• **Z-Direction:** Finally, the \( z \) value spans from 0 to 1, representing a vertical block from floor to ceiling within the defined \( x \) and \( y \) limits.

Understanding this region helps you visualize where the integration is supposed to happen. Often, visualizing or sketching this region can make the process of integration and the choice of order of integration more intuitive.

The concept of a definite integral allows us to calculate the accumulated value of a function within specified bounds. In triple integrals, this principle extends to three dimensions, capturing volume-like accumulations. Here, the integral to be solved is:\[ \int_0^1 \int_{-x^2}^{x^2} \int_0^1 (xy + yz + xz) \, dz \, dy \, dx. \]
Each specific integral boundary defines the range over which the function \( xy + yz + xz \) is evaluated:

• The innermost integral evaluates \( z \) from 0 to 1.
• The middle integral applies to \( y \), ranging from \( -x^2 \) to \( x^2 \).
• The outermost integral considers \( x \) from 0 to 1.

The step-by-step integration is based on evaluating these boundaries in order. The calculated result, \( \frac{4}{15} \), provides the total value of the function summed over all these mini volumes, giving insights into the overall characteristics of the specified region. This total volume (or accumulated value) is the heart of what definite integrals aim to uncover.

Integration with respect to variables is a key technique in solving multivariable integrals. Here, the function \( xy + yz + xz \) is integrated one variable at a time, cycling through \( z, y, \) and finally \( x \). Each integration simplifies the expression further, peeling back layers of the function like an onion.

Step-by-step Integration

1. **Integrate with Respect to \( z \):** We evaluate the integral \[\int_0^1 (xy + yz + xz) \, dz = xy + \frac{y}{2} + \frac{x}{2}.\] This involves treating \( xy \), \( yz, \) and \( xz \) as functions of \( z \) alone, integrating each term separately, then plugging in the bounds for \( z \).

• The results yield expressions in \( x \) and \( y \).

2. **Integrate with Respect to \( y \):** Using the result from \( z \)'s integration, evaluate for \( y \):
\[ \int_{-x^2}^{x^2} \left( xy + \frac{y}{2} + \frac{x}{2} \right) \, dy. \]

• This calculation results in a function solely depending on \( x \).

3. **Integrate with Respect to \( x \):** Finally, integrate across \( x \):
\[ \int_0^1 \left( x^5 + \frac{x^4}{2} \right) \, dx, \]resulting in the combined essence of all three dimensions.Concluding this sequential integration comprehensively covers the multivariable function, showcasing how constraints on each variable alter the outcome in cumulative steps.