 Suppose we want to find the area of a region. In single variable calculus, we can sum the areas of rectangles. In multivariable calculus, each rectangle can be the sum of squares. Geometrically, this looks like the following. Suppose our element of area is a square dx-wide by dy-high. We can sum the squares from y equals a to y equals b. And this gives us a rectangle. We can then sum the rectangles from x equals c to x equals d. And this gives us the area as a iterated integral. Now to set that up, remember the differential variable is controlling. So our first sum changed y and went from y equals a to y equals b. Meanwhile, our second sum changed x and went from x equals c to x equals d. For example, let's express the area of the region shown as two different iterated integrals. So we can take our element of area. Now if we sum by letting x vary first moving horizontally, then we see that x goes from x equals negative 1 to x equals 3 and y goes from y equals 3 to y equals 8. And so we can write our integral. First we'll sum letting x vary from x equals negative 1 to x equals 3 and then we'll sum letting y vary from y equals 3 to y equals 8. We could also sum by letting y vary first and so we'll move vertically and so in that case y goes from y equals 3 to y equals 8 and then x will vary from x equals negative 1 to x equals 3. And so we can write our integral as the sum when we let y vary from 3 to 8 then let x vary from negative 1 to 3. So what if our region isn't a rectangle? This there's some strategies that are useful to keep in mind. We'll try to avoid using the same curve more than once. Now sometimes we have to and we have no choice but it's easier if we make the effort to not use the same curve twice and then the other thing to keep in mind always is the differential variable is the variable that changes. For example let's try to write a double integral to find the area of the region shown. And the first thing to notice is that if we move our differential square vertically in some parts we'll encounter two sections of the same curve and in other parts we'll encounter two different curves. And this makes things hard. We might have to do this but let's see if we can avoid it. And notice that if we move our differential square horizontally we're always using the same boundary curves. On the left hand side it's always the parabola and on the right hand side it's always the straight line. And this makes it easier. So our first integrand will move our differential square horizontally and since the x values are changing our differential will be dx and our limits must also be values of x. Now our least values are going to be on the curve x equals y minus 2 squared and since that's already in terms of x we can use that as our lower limit. Our greatest x values are going to be on the straight line 3x plus 4y equals 12. Now we need to express this as a value of x so let's solve for x and that gives us our upper limit. Now if we let x vary between these two limits this will give us a rectangle and moving it vertically means our differential variable is y and our greatest and least values will be at the intersection points. And so we solve to find those intersection points and we see that our y values go from 0 to 8 thirds.