Multiple integrals
👋 About the lesson
Welcome to the last exercise session of this semester! I see this week as a preparation what you can expect and will use in the upcoming Applied Statistics course which covers fundamentals of probability theory and statistics.
We will specifically talk about how to compute double integrals over some given area. This is useful when you for instance want to compute joint probability distribution of two random variables. Without further ado, let’s get started.
📓 Notes
Double integrals (how to compute and interpret)
We know from the 10th lecture how to compute integral over given interval \((a, b)\) for given function \(f(x)\). Just a short recap, consider the following integral:
\[\int_1^3 x^2 dx\]To compute its value, we first find the antiderivative function \(F(x)\) to the original function \(f(x)\):
We can the compute the integral value as:
\[\int_1^3 x^2 dx = F(3) - F(1) = 9 - \frac{1}{3}\]In addition, recall that the obtained value can be interpretted as an area under the function within the given interval.
Now, let’s consider a double integral example:
\[\int_1^2 \int_0^3 x^2 + y dxdy\]Notice in addition to \(x\), we now also integrate over \(y\), i.e., we integrate over a multivariate function \(f(x, y) = x^2 + y\). Further, it is important to consider the order of \(dx\) and \(dy\) since it determines with respect to which variable we are integratting the inner and outer integral. Further, notice that we integrating over a rectangle which is defined from 0 to 3 on x-axis and and 1 to 2 on the y-axis. We first by integrating the inner intgral with respect to \(x\) (i.e., we treat \(y\) as a constant). We again start by finding the antiderivative:
Therefore:
\[\int_1^2 F(3) - F(0) dy = \int_1^2 9 + 3y dy\]Using similar procedure for the outer integral we first find the antiderivative:
And finally we obtain:
\[\int_1^2 9 + 3y dy = F(2) - F(1) = 18 + 6 - (9 + \frac{3}{2}) = 15 - \frac{3}{2}\]Quite easy right? Essentially nothing new except more variables, but you still focus on one at a time. I suggest you check this Geogebra applet to get also visual idea of what is going on. As you should be able to see, when you keep the given dimension constant, it graphically means that you are integrating over single variate function with given interval. (inner integral) You then need to sum these 1D slices together using the outer integral.
So far we have integrated over the rectangular areas, however, what if the shape is more complex, i.e., its boundary is defined by a function? See the exercise 14 for couple examples. (ex. 15.2.31 and 15.4.29) There, I also show how to use the double integral to compute probabilities for pairs of indepenedent random variables.
⛳️ Learning goals checklist
This week has been fairly straightforward, you should be able to:
- compute double integrals over rectangles and areas bounded by functions (type I and II in the book)
- compute probabilities for pairs of independent random variables
I will be grateful if you could write me short feedback about what you liked/disliked about my TA sessions. (takes max 3 min) Lastly, thank you for the whole semester, and good luck at the exam!