Vector Calculus: Area and Mass of a Region

[Kushwoho44]

Hi y'all.

Here is exactly what is stated on the theory page of my book:

Example: Area of a Region
The area of a region R in the xy-plane corresponds to the case where f(x,y)=1.

Area of R= ∫∫dR

Example: Mass of a Region
The mass of a region R in the xy-plane with mass density per unit area ρ(x,y) is given by:

Mass of R= ∫∫ρdR

I'm not at all understanding this first part of the theory, why is it that the area of the region R in the xy-plane is the case f(x,y)=1 and how did they obtain that express for the area R?

All help is immensely appreciated as I'm tearing my hair out over this.Thanks!

 

[HallsofIvy]

What do you understand about integrals? Most people are first introduced to integrals in the for $\int_a^b f(x)dx$ where it is defined as "the area of the region bounded by the graphs of y= f(x), y= 0, x= a, and x= b".

Once you start dealing with double integrals, since $f(x)= \int_0^{f(x)} dy$, it is easy to see that we can write that integral, and so that area, as $\int_a^b\int_0^{f(x)} dy dx$. From that we can see that "dxdy" acts as a "differential of area". That is, we find the area of any region by integrating $\int\int dx dy$ over that region. Similarly, in three dimensions, we can find the volume of a region by integrating $\int\int\int dxdydz$ over that region.

Another way of reaching the same idea is to divide the region into small rectangles with sides parallel to the x and y axes and identifying the lengths of the sides as "$\Delta x$" and "$\Delta y$". Of course, the area of each such rectangle is $\Delta x\Delta y$ and the area of the whole region can be approximated by $\sum \Delta x\Delta y$. "Approximate" because some of the region, near the bounds, will not fit neatly into those rectangles. But we can make it exact by taking the limit as the size of $\Delta x$ and $\Delta y$ go to 0. Of course, I have no idea what method your texts or courses used to introduce the double integral so I cannot be more precise.

 

[Kushwoho44]

Thanks for that, the stupid part I wasn't understanding was that f(x,y) maps onto z.

Makes perfect sense now.