Observe that the inside integral cannot be expressed in terms of elementary functions.change the order of integration and so evaluate the double integral. $$ \int_{x=0}^{2} \int_{y=x}^{2} e^{-y^{2} / 2} d y d x $$

The order of integration in a double integral determines which variable you integrate first.
In the original problem, we start by integrating with respect to y, followed by x. But, it's sometimes easier to switch this order.

When sketching the region of integration, note the bounds provided:

• x ranges from 0 to 2
• y ranges from x to 2

By changing the order of integration, we swap the variables and adjust the bounds accordingly. For our new integral:

• y ranges from 0 to 2
• x ranges from 0 to y

The rewritten integral becomes:
\(\begin{equation*} \begin{aligned} \ \int_{y=0}^{2} \int_{x=0}^{y} e^{-y^{2} / 2} \, dx \, dy \ \end{aligned} \ \end{equation*}\). Swapping the order of integration often simplifies the computation process. Make sure to always adjust the integration limits properly!

U-substitution is a method used to simplify the process of integration by substituting a part of the integral with a new variable.
After simplifying the inner integral, \(\begin{equation*} ye^{-y^2/2} \end{equation*}\), we must integrate it with respect to y.

Consider using substitution to handle the exponent with ease. Let's set \(\begin{equation*} u = -\frac{y^2}{2} \ \)\. Then, take the derivative: \(\begin{equation*} du = -y \, dy \end{equation*}\). Rewrite the integral using u:
\(\begin{equation*} \int_{y=0}^{2} y \, e^{-y^2/2} \, dy = - \, \int_{0}^{-2} e^{u} \ du \end{equation*}\).

This transformation allows us to deal with a more straightforward integral involving the exponential function. Remember to revert back to the original variable once you've solved the integral.

Understanding the region of integration is crucial when working with double integrals. The region of integration represents the domain over which you perform the integrations.
For the given problem, the region is described by the bounds of x and y.

Initially, the bounds are:

• x ranges from 0 to 2
• y ranges from x to 2

To change the order of integration, we sketch this region.
This gives a visual understanding of the limits:

• y now ranging from 0 to 2
• x from 0 to y

Visualizing the region helps to correctly set the limits after changing the integration order. Use graph paper or a software tool to sketch these regions whenever you find yourself in doubt!

In some cases, integration by parts can be required. This method helps solve integrals involving products of functions. However, for this specific exercise, we do not need it.
Recall the formula for integration by parts: \(\begin{equation*} \int u \, dv = uv - \int v \, du, \end{equation*}\).

Choose \(\begin{equation*} u \) and \(\begin{equation*} dv \) appropriately.
For our problem, we managed with u-substitution, which avoided complex parts of the integration by parts.

Always determine the easiest method applicable. Use integration by parts when other strategies do not simplify the integral adequately.