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Let's break down and solve this triple integral step-by-step.

### Given Integral:
\[
\int_0^1 \int_0^{1-x} \int_y^2 dz \, dy \, dx
\]

1. **Innermost Integral**:
   \[
   \int_y^2 dz
   \]
   This is a simple integral with respect to \(z\). Since the integral of \(dz\) is simply \(z\), we get:
   \[
   \int_y^2 dz = \left[z\right]_y^2 = 2 - y
   \]

2. **Middle Integral**:
   Now, we integrate with respect to \(y\) from 0 to \(1-x\):
   \[
   \int_0^{1-x} (2 - y) \, dy
   \]
   Let's break this into two separate integrals:
   \[
   \int_0^{1-x} 2 \, dy - \int_0^{1-x} y \, dy
   \]
   The first integral is:
   \[
   \int_0^{1-x} 2 \, dy = 2y \bigg|_0^{1-x} = 2(1-x)
   \]
   The second integral is:
   \[
   \int_0^{1-x} y \, dy = \frac{y^2}{2} \bigg|_0^{1-x} = \frac{(1-x)^2}{2}
   \]
   Putting these together, we have:
   \[
   \int_0^{1-x} (2 - y) \, dy = 2(1-x) - \frac{(1-x)^2}{2}
   \]

3. **Outermost Integral**:
   Finally, we integrate with respect to \(x\) from 0 to 1:
   \[
   \int_0^1 \left[ 2(1-x) - \frac{(1-x)^2}{2} \right] dx
   \]
   Let's simplify the expression inside the integral:
   \[
   2(1-x) - \frac{(1-x)^2}{2} = 2 - 2x - \frac{1 - 2x + x^2}{2}
   \]
   Expand and simplify:
   \[
   2 - 2x - \frac{1}{2} + x - \frac{x^2}{2} = \frac{3}{2} - x - \frac{x^2}{2}
   \]
   
4. **Integrating the Final Expression**:
   \[
   \int_0^1 \left( \frac{3}{2} - x - \frac{x^2}{2} \right) dx
   \]
   We split this into separate integrals:
   \[
   \int_0^1 \frac{3}{2} \, dx - \int_0^1 x \, dx - \int_0^1 \frac{x^2}{2} \, dx
   \]
   The first integral is:
   \[
   \int_0^1 \frac{3}{2} \, dx = \frac{3}{2}x \bigg|_0^1 = \frac{3}{2}
   \]
   The second integral is:
   \[
   \int_0^1 x \, dx = \frac{x^2}{2} \bigg|_0^1 = \frac{1}{2}
   \]
   The third integral is:
   \[
   \int_0^1 \frac{x^2}{2} \, dx = \frac{x^3}{6} \bigg|_0^1 = \frac{1}{6}
   \]
   
5. **Putting It All Together**:
   \[
   \frac{3}{2} - \frac{1}{2} - \frac{1}{6} = \frac{3}{2} - \frac{4}{6} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}
   \]

### Final Answer:
\[
\frac{5}{6}
\]

Would you like further details or have any questions? Here are some related questions that expand on this problem:

1. How would the result change if the limits of integration for \(y\) and \(z\) were different?
2. What is the geometric interpretation of the region of integration?
3. How do we change the order of integration for a triple integral?
4. How would you approach this problem using a numerical integration method?
5. What if the integrand were more complex, such as a polynomial in \(z\), \(y\), and \(x\)?

**Tip**: When dealing with nested integrals, always start by solving the innermost integral and work outward!

Solve the triple integral: ∫_0^1 ∫_0^(1-x) ∫_y^2 dz dy dx

Step-by-step solution to the triple integral ∫_0^1 ∫_0^(1-x) ∫_y^2 dz dy dx. This solution covers key multivariable calculus concepts and involves breaking down the integrals in stages. Suitable for undergraduate students studying multivariable calculus.

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