Order of integration help

[AZCalcstudent]

Hey internet, I'm having trouble with one of the problems on a maple assignment I have, maybe someone here can get it.

Verify Fubini’s Theorem by evaluating each of the double integrals below in each of the two possible orders

2. ∫∫_RdA where R is the region enclosed by the curves y = -x²+1 and y = x²-1.

I got it ok doing it in the order of dydx, where x varies from -1 to 1, y varies from x²-1 to -x²+1, but I can't figure out how to accurately switch it to dxdy

 

[tkhunny]

You may wish to consider some symmetry:

\(\displaystyle 4\cdot \int_{0}^{1}\int_{0}^{\sqrt{1-y}}1\;dx\;dy\)

If math text ever works, you will see a focus on ONLY the first quadrant.

 

[galactus]

If math text ever works

I sent Ted an email, but no response. I am sure others have as well. He must be out of town.

 

[HallsofIvy]

The highest point on the graph of y= 1- x^2 is (0, 1) and the lowest point on the graph of y= x^2- 1 is (0, -1) so to integrate using the order "dxdy" y must range from -1 to 1. However, the range of x for each y changes at y= 0 so you will need to have two integrals, one with y going from -1 to 0, the other from 0 to 1. For y from -1 to 0, the "sides" are given by y= x^2- 1. Find the limits on the dx integral by solving that for x. For y from 0 to 1, the "sides" are given by y= 1- x^2. Find the limits on the dx integral by solving that for x.

 

[AZCalcstudent]

Thanks for the replies. I ended up using the 4 * first quadrant method; have yet to get it back graded.