# Vector Calculus: Area and Mass of a Region

by Kushwoho44
Tags: calculus, mass, region, vector
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,542 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.