Sketch the region enclosed by x+y2=12x+y2=12 and x+y=0x+y=0. decide whether to integrate with respect to xx or yy, and then find the area of the region.

Step-by-step explanation:

For this case we have these two functions:

  (1)

  (2)

And as we can see we have the figure attached.

For this case we select the x axis in order to calculate the area.

If we solve y from equation (1) and (2) we got:

Now we can solve for the intersection points:

And the solutions are

So then we have in total 3 intersection point

And we can find the area between the two curves separating the total area like this:

We can separate the integrals like this:

For this integral we can use the u substitution with and after apply and solve the integral we got:

The other integral:

And for the other integral:

We can use the same substitution and after replace and solve the integral we got:

So then the final area would be given adding the 3 results as following: