Question

Evaluate the double integral. Note that it is necessary to change the order of integration. $$ \int_{0}^{2} \int_{x}^{2} e^{-y^{2}} d y d x $$

Short answer

The solution to the double integral is \(-\frac{1}{2} e^{-4} + \frac{1}{2}\) after changing the order of integration.

Step-by-step solution

Understand the given integral

The given integral \(\int_{0}^{2} \int_{x}^{2} e^{-y^{2}} d y d x \) has its limits of inner integral as \(x\) and \(2\). This indicates that the region of integration is bounded by \(y=x\) and \(y=2\), and \(x\) varies from \(0\) to \(2\).

Change the order of integration

In order to change the order of integration, the limits need to be adjusted accordingly. The outer integral will now be with respect to \(y\) and the inner one with respect to \(x\). The limits for \(y\) will be from \(0\) to \(2\) and for \(x\) will be from \(0\) to \(y\). This gives us the integral: \(\int_{0}^{2} \int_{0}^{y} e^{-y^{2}} d x d y\).

Evaluate the inner integral

The inner integral \(\int_{0}^{y} e^{-y^{2}} d x\) is independent of \(x\), so \(x\) is just a multiplier. Therefore, the integral becomes \(x e^{-y^2}\) evaluated from \(0\) to \(y\), which simplifies to \(y e^{-y^2}\). The original double integral can now be written as \(\int_{0}^{2} y e^{-y^2} d y\).

Evaluate the outer integral

The outer integral is now a standard form of integral that can be solved by substitution method. Let \(u = -y^2\), then \(du = -2y dy\). With these substitutions, our integral becomes \(-\frac{1}{2} \int e^u du\), which simplifies to \(-\frac{1}{2} e^u\). Substituting back the original variables, we get \(-\frac{1}{2} e^{-y^2}\) evaluated from \(0\) to \(2\).

Final calculation

The value after performing the final calculation is \(-\frac{1}{2} e^{-4} + \frac{1}{2}\). This is the result of the given double integral after changing the order of integration.