Changing order of integration is not always feasible

Suppose you need to calculate the double integral $\iint_{D}^{}$f (x, y) dA for some function f (x, y) and the region D shown below.

To calculate the double integral, you can write it as an iterated integral. For example, let's say that in the region D, the lowest value of y is a and the highest value of y is b. In other words, the range of y in the region D is a$\le$y$\le$b.

For a given value of y, the range of x in D depends on the value of y. However, the region is nice enough so that the range of x for any y is just a simple interval. We could define two functions h₁(y) and to h₂(y) so that this interval in x is [h₁(y), h₂(y)] for each value of y. This description of the region D is shown in the following picture.

Since the region D is defined by

$$\le \le$$

$$\le \le$$

we can represent double integral of f (x, y) over D as the following iterated integral,

$$\iint_{D}^{} \int_{a}^{b} \left(\vphantom{ \int_{h_1(y)}^{h_2(y)} f(x,y) dx }\right. \int_{{h_1(y)}}^{{h_2(y)}} \left.\vphantom{ \int_{h_1(y)}^{h_2(y)} f(x,y) dx }\right)$$

However, we would run into trouble if we tried to change the order of integration. The difficulty is that the range of y for some values of x is not a simple interval. For example, for the value of x given by the vertical dashed line below, the range of y is two different intervals.

Since we cannot always write the range of y as a single interval [g₁(x), g₂(x)], we cannot write the integral $\iint_{D}^{}$f (x, y) dA as a single iterated integral of the form

$$\iint_{D}^{} \int_{c}^{d} \left(\vphantom{ \int_{g_1(x)}^{g_2(x)} f(x,y) dy }\right. \int_{{g_1(x)}}^{{g_2(x)}} \left.\vphantom{ \int_{g_1(x)}^{g_2(x)} f(x,y) dy }\right)$$

If we really wanted to integrate with respect to y first, we'd have to break the region D into pieces and compute separate integrals for each piece. We'll leave that procedure to your imagination.