# How can I calculate this region area

How can I calculate a region area using a double integral e.g. $D=[a,b]\times[c,d]$. Do I use the below formula? If Yes how?

$$\iint_D f(x,y)\,dx\,dy\:=\int _a^b\left(\int _c^df\left(x,y\right)\,dy\right)dx$$

Also what if I have $D$ something like $D=[1,3]\times[0,2]$.

Thanks

• Typically you integrate the constant function $f(x, y) = 1$ over the region to find its area; for the limits on the integral, you use $a = 1, b = 3, c = 0, d = 2$. – John Hughes Dec 28 '16 at 23:22

$$\iint_D f(x,y)\,dx\,dy\:=\int _a^b\left(\int _c^df\left(x,y\right)\,dy\right)dx$$
Take $f(x,y)=1$, $a=1, b=3, c=0, d=2$.
Area$=\int_1^3dx\int_0^2dy=(3-1)(2-0)=4$
The trick to multiple integrals is to look at the region you're integrating over. Here, our D tells us that x goes from 1 to 3 and y goes from 0 to 2 and this region is a rectangle. Then you plug in what you know and make sure the bounds match; for example the integral should be $\int_{0}^{2}\int_{1}^{3}f(x,y)\text{d}x\text{d}y$