Interpretation of the double integral over a region

Question

I am studying multivariable calculus, and I came up with a doub that I cannot solve, or I believe I udnerstood wrong.

Say I have to calculate the following integral:

$$\int_T (x^2 + y)\ \text{d}x\text{d}y$$

Where $T$ is a well defined region (for example $T = \{(x, y)\in\mathbb{R}^2; 0\leq x \leq 1,\ 1\leq y \leq 2\}$).

Now, I understood the double integration is mostly used to calculate areas of given regions or shapes. For example, the above integral alone, that is

$$\int_T \text{d}x\text{d}y$$

Will give me the area of the region $T$, which by the way is a square. So fine so far.

My question is: what does exactly mean to calculate the integral of $x^2 + y$ over this region? $x^2 + y$ defines a region too, a shape (right?), yet not exactly since I expect a shape to be defined by a function $f(x, y) = 0$, not just $f(x, y)$.

I know it may be a stupid question, but I need to get those suble things to touch what I am studying, and my notes are not that useful. Thank you!

My biggest suggestion to you is to not get bogged down by one interpretation of an integral. There are several possible ways to interpret the numerical quantity obtained by integrating a function over a region. See Visualization of double integrals for more details. — peek-a-boo, Jun 11 '22 at 17:51

score 1 · Answer 1 · answered Jun 11 '22 at 20:10

The integral I of x^2+y can only be thought of related to a region (other than T) by considering a set in 3D: in the simple case of a positive function (which is true in your situation) you get the volume of the set of points (x,y,z) such that (x,y) belongs to T and 0 <= z <= the integrand, i.e. 0 <= z <= x^2+y in your case. Forget your other considerations

Interpretation of the double integral over a region

1 Answers1