It looks like you want to change the order of integration for the double integral $\int \int x \, \text{He} \, dx \, dy$, where the given limits for $x$ are from $y$ to 1 and for $y$ are from 0 to 1. There seems to be a part of the integrand ($\text{He}$) that's unclear or might be a typo.

I'll assume that the integrand is simply $x$ and the term $\text{He}$ might not be part of the mathematical expression. Let's proceed under that assumption and change the order of integration.

Original Integral:

$I = \int_0^1 \left( \int_y^1 x \, dx \right) dy$

Step 1: Change the order of integration

To change the order of integration, we need to carefully determine the new limits. The region of integration is described as:

• $x$ goes from $y$ to 1.
• $y$ goes from 0 to 1.

This means the region is a triangular area on the $x$-$y$ plane, bounded by $x = y$ and $x = 1$.

To reverse the order of integration, we need to describe the region by:

• $x$ goes from 0 to 1.
• For each fixed $x$, $y$ ranges from 0 to $x$.

Thus, the new order of integration is:

$I = \int_0^1 \left( \int_0^x x \, dy \right) dx$

Step 2: Evaluate the integral

Now, let's evaluate the integral:

$I = \int_0^1 \left( \int_0^x x \, dy \right) dx$

1. First, evaluate the inner integral with respect to $y$:

$\int_0^x x \, dy = x \cdot (x - 0) = x^2$

2. Now, substitute this into the outer integral:

$I = \int_0^1 x^2 \, dx$

3. Evaluate the outer integral:

$I = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}$

Final Answer:

$I = \frac{1}{3}$

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Would you like further details on this calculation? Feel free to ask if you need more clarification or if there are other concepts you'd like to explore!

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Related Questions:

1. How would the integral change if the limits for $x$ and $y$ were reversed?
2. What happens if the integrand is a function other than $x$ (e.g., $x^2$ or $\sin(x)$)?
3. How do you apply cylindrical coordinates in a triple integral?
4. What is the significance of changing the order of integration in double integrals?
5. How do you approach integrals in polar coordinates when the region is circular?

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Tip: When changing the order of integration, always visualize the region of integration on a graph to ensure you correctly adjust the limits. This step can help prevent mistakes when switching the order!