3. (15 pts) Calculate the iterated integral by changing the order of integration.
\[
\int_{0}^{1} \int_{4 x}^{4} e^{-y^{2}} d y d x
\]

Step 1 ：The given integral is a double integral with the inner integral with respect to y and the outer integral with respect to x. The limits of the inner integral are functions of x, and the limits of the outer integral are constants.

Step 2 ：To change the order of integration, we need to express the limits of the integrals in terms of y. The region of integration is defined by \(0 \leq x \leq 1\) and \(4x \leq y \leq 4\). If we express x in terms of y, we get \(x = y/4\).

Step 3 ：The new limits for y are obtained by substituting \(x = 0\) and \(x = 1\) into the equation \(y = 4x\), which gives \(y = 0\) and \(y = 4\).

Step 4 ：So, the new limits of integration are \(0 \leq y \leq 4\) and \(0 \leq x \leq y/4\). The iterated integral with the order of integration changed is then: \[\int_{0}^{4} \int_{0}^{y/4} e^{-y^{2}} d x d y\]

Step 5 ：Calculate the inner integral first, which gives \(y*exp(-y^2)/4\).

Step 6 ：Then calculate the outer integral, which gives \(1/8 - exp(-16)/8\).

Step 7 ：Final Answer: The value of the iterated integral after changing the order of integration is \(\boxed{\frac{1}{8} - \frac{e^{-16}}{8}}\).