Reverse the order of integration in the following integral. \[ \int_{0}^{1} \int_{4}^{4 e^{x}} f(x, y) d y d x \] Reverse the order of integration. (Type exact answers.)

Step 1 ：Understand the limits of the original integral. The inner integral is with respect to y and ranges from 4 to 4e^x. The outer integral is with respect to x and ranges from 0 to 1.

Step 2 ：To reverse the order of integration, we need to express x in terms of y and find the new limits for y and x.

Step 3 ：From the limits of the inner integral, we can see that y = 4e^x. Solving for x, we get x = ln(y/4).

Step 4 ：The lower limit for y is the minimum value of 4e^x, which is 4 when x = 0. The upper limit for y is the maximum value of 4e^x, which is 4e when x = 1.

Step 5 ：The lower limit for x is the minimum value of ln(y/4), which is 0 when y = 4. The upper limit for x is the maximum value of ln(y/4), which is 1 when y = 4e.

Step 6 ：So, the reversed integral is \(\int_{4}^{4e} \int_{0}^{\ln(y/4)} f(x, y) dx dy\)

Step 7 ：Final Answer: \(\boxed{\int_{4}^{4e} \int_{0}^{\ln(y/4)} f(x, y) dx dy}\)