91Ó°ÊÓ

When dealing with double integrals, the order of integration indicates which variable we integrate with respect to first. In the original problem, we have the integral \[\int_{0}^{1} \int_{y}^{1} x^{2} e^{x y} \, dx \, dy\]which means that we integrate with respect to \(x\) first, and then \(y\). These limits suggest that for each fixed value of \(y\), \(x\) ranges from \(y\) to \(1\). Finally, \(y\) ranges from \(0\) to \(1\).
Reversing the order means changing which variable we integrate first and adjusting the corresponding limits of integration. This often simplifies the integration process. In this exercise, we change the variables to have the integration with respect to \(y\) first resulting in:
\[\int_{0}^{1} \int_{0}^{x} x^{2} e^{x y} \, dy \, dx\]Now, for a fixed \(x\), \(y\) varies from \(0\) to \(x\). This switch clarifies the region and sometimes makes calculations easier.

After deciding on the order of integration, the next step is to evaluate the integrals. In our problem, we first evaluate the inner integral \( \int_{0}^{x} x^{2} e^{xy} \, dy \). Here, because we have two variables \(x\) and \(y\), but are integrating first with respect to \(y\), \(x\) will be treated as a constant. Evaluating this integral requires attention to the bounds and the function itself.
After evaluating the inner integral, substitute back its result into the outer integral to get something like:\[\int_{0}^{1} x^2 \left( \frac{1}{x} (e^{x^2} - 1) \right) \ dx = \int_{0}^{1} x (e^{x^2} - 1) \ dx\]Separate this into manageable parts by distributing and splitting the original integral into more straightforward components:
\[\int_{0}^{1} x e^{x^2} \ dx - \int_{0}^{1} x \ dx\]Allow this decomposition to ease the process of calculating the definite integral.

U-substitution is a technique for simplifying the integration process, particularly for complex functions. It acts like the reverse of the chain rule in differentiation. This method is useful when a part of the integrand is recognized as the derivative of another part. In our calculation, we used substitution to evaluate \(\int_{0}^{1} x e^{x^2} \, dx\).
Begin by letting \(v = x^2\), then \(dv = 2x \, dx\) or \(x \, dx = \frac{1}{2} \, dv\). This substitution changes the integral into a simpler form:\[\frac{1}{2} \int_{0}^{1} e^{v} \, dv\]Now you can integrate \(e^v\) easily because it's a straightforward exponential function. Substitute back after integrating, which yields:\[\frac{1}{2} (e^{1} - e^{0}) = \frac{1}{2} (e - 1)\]U-substitution reduces complexity and helps in tackling challenging integrals effectively.

Understanding the region of integration is crucial because it determines the limits for the variables. In a double integral, this region is a subset of the plane. For our problem, we have initial limits indicating we are integrating over the region described by \(y = 0\) to \(y = 1\) and \(x = y\) to \(x = 1\).
This region is a triangle beneath the line \(x = y\). The boundaries form a right triangle with vertices at points (0, 0), (1, 0), and (1, 1). Sketching this area aids in visualizing how the variable limits change when reversing the order of integration. After reversing, \(x\) is integrated first from 0 to 1, and \(y\) from 0 to \(x\).
Such comprehension of the region helps in correctly setting and modifying the limits when the order of integration is reversed, ensuring the integral is still effectively representing the same region.