Integrating multivariate densities with variable bounds.

Question

I am struggling how to deal with certain bounds when it comes to integrating multivariate densities.

How would you specify the bounds to integrate a function such as $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y) \,dx\,dy$ with bounds like

(i) $0<x<y<2$

(ii) $0<x<1$, $0<y<\sqrt(x)$

I don't know whether these examples make sense, I just made them up to go one step further from what we did in class. I know how to deal with for example $0<x$, $0<y$, $x+y<1$, but as soon as the bounds get more complex I fail to understand how to approach it. Is there a general method to go about more complex bounds as I have outlined above?

Answer 1

To be precise, what you mean is to find a double integral of the form $$ \iint_D f(x,y)dxdy $$ where $D$ is a region defined by inequalities or to find $$ \iint_{\mathbb{R}^2} f(x,y)dxdy $$ where $f$ is "supported" on the region $D$.

For those integrals that can be calculated by hand, your goal is to write it as iterated integrals. To figure out the bounds, drawing a picture would be helpful. See below for your examples.

How to find the bounds should be a very standard materials you see in a textbook when you learn double integrals the first time.

In general, the "easy" integrals can be written as one of the following ways: $$ \int_a^b\left(\int_{c(x)}^{d(x)}f(x,y)dy\right)dy,\quad \int_a^b\left(\int_{c(y)}^{d(y)}f(x,y)dx\right)dy \tag{1} $$

So if the region $ D$ has been written in one of the following forms: $$ D=\{(x,y): c(x)<y<d(x), a<x<b\},\quad D=\{(x,y): c(y)<x<d(y), a<y<b\} $$ you should be happy since it tells you directly the bounds in (1).

Your third example, $$ x>0,\quad y>0,\quad x+y<1 $$ can be written as $$ D=\{(x,y): 0<y<1-x, 0<x<1\} $$ or $$ D=\{(x,y): 0<y<1, 0<x<1-y\} $$

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Answer 2

Presumably you are trying to handle $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y) \,dx\,dy$ where you know $f(x,y)=0$ outside the bounds. A sensible way to approach this is with indicator functions, but sometimes the answer is sufficiently clear anyway.

In your two particular examples:

(i) $0<x<y<2$ can be $$\int_{y=0}^{2}\int_{x=0}^{y} f(x,y) \,dx\,dx$$ or equivalently $$\int_{x=0}^{2}\int_{y=x}^{2} f(x,y) \,dy\,dx$$

(ii) $0<x<1$, $0<y<\sqrt x$ can be $$\int_{x=0}^{1}\int_{y=0}^{\sqrt x} f(x,y) \,dy\,dx$$