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We are given the triple integral: $\int_0^1 \int_0^{5-5x} \int_0^{5-5x-y} dz \, dy \, dx$

Step-by-step Solution:

1. Evaluate the innermost integral: $\int_0^{5-5x-y} dz = [z]_0^{5-5x-y} = (5 - 5x - y) - 0 = 5 - 5x - y$

   After solving this, the integral becomes: $\int_0^1 \int_0^{5-5x} (5 - 5x - y) \, dy \, dx$

2. Evaluate the second integral (in terms of $y$): Expand the expression $5 - 5x - y$: $\int_0^{5-5x} (5 - 5x - y) \, dy = \int_0^{5-5x} (5 - 5x) \, dy - \int_0^{5-5x} y \, dy$

   Evaluate each term:

   • For $\int_0^{5-5x} (5 - 5x) \, dy$: $\int_0^{5-5x} (5 - 5x) \, dy = (5 - 5x) \cdot y \Big|_0^{5-5x} = (5 - 5x)(5 - 5x) = (5 - 5x)^2$

   • For $\int_0^{5-5x} y \, dy$: $\int_0^{5-5x} y \, dy = \frac{y^2}{2} \Big|_0^{5-5x} = \frac{(5 - 5x)^2}{2}$

   Combine the results: $\int_0^{5-5x} (5 - 5x - y) \, dy = (5 - 5x)^2 - \frac{(5 - 5x)^2}{2} = \frac{(5 - 5x)^2}{2}$

   So the integral becomes: $\int_0^1 \frac{(5 - 5x)^2}{2} \, dx$

3. Evaluate the outermost integral (in terms of $x$): Expand $(5 - 5x)^2$: $(5 - 5x)^2 = 25 - 50x + 25x^2$

   Substitute into the integral: $\int_0^1 \frac{(5 - 5x)^2}{2} \, dx = \frac{1}{2} \int_0^1 (25 - 50x + 25x^2) \, dx$

   Evaluate each term:

   • For $\int_0^1 25 \, dx$: $\int_0^1 25 \, dx = 25x \Big|_0^1 = 25(1) - 25(0) = 25$
   • For $\int_0^1 -50x \, dx$: $\int_0^1 -50x \, dx = -50 \cdot \frac{x^2}{2} \Big|_0^1 = -50 \cdot \frac{1^2}{2} = -25$
   • For $\int_0^1 25x^2 \, dx$: $\int_0^1 25x^2 \, dx = 25 \cdot \frac{x^3}{3} \Big|_0^1 = 25 \cdot \frac{1^3}{3} = \frac{25}{3}$

   Combine the results: $\int_0^1 (25 - 50x + 25x^2) \, dx = 25 - 25 + \frac{25}{3} = \frac{25}{3}$

   Multiply by $\frac{1}{2}$: $\int_0^1 \frac{(5 - 5x)^2}{2} \, dx = \frac{1}{2} \cdot \frac{25}{3} = \frac{25}{6}$

Final Answer:

$\int_0^1 \int_0^{5-5x} \int_0^{5-5x-y} dz \, dy \, dx = \frac{25}{6}$

Would you like a deeper explanation of any step?

Follow-up Questions:

1. How do we interpret triple integrals geometrically?
2. What does the region of integration look like in this example?
3. Can the order of integration be changed for this problem?
4. How would we evaluate this integral using numerical methods?
5. What are common applications of triple integrals in physics or engineering?

Tip:

Always check the bounds of integration to ensure they describe a valid geometric region before evaluating a multiple integral!