Double Integrals: Limits Explained

[coverband]

On my post titled double integrals you explained why limits go from x^2 to x when integrating wrt y (i.e. the "bottom" graph is the bottom limit) however i seem to have found a webpage that disagrees with you http://www.libraryofmath.com/double-integral-over-a-more-general-region.html example (d) has limits in opposite direction you stated.

 

[NoMoreExams]

On my post titled double integrals you explained why limits go from x^2 to x when integrating wrt y (i.e. the "bottom" graph is the bottom limit) however i seem to have found a webpage that disagrees with you http://www.libraryofmath.com/double-integral-over-a-more-general-region.html example (d) has limits in opposite direction you stated.

It doesn't disagree, you just have to consider 2 different regions there since y = x - 1 isn't always below y^2 = 2x + 6. In fact on the interval [-3, -1], y = x - 1 doesn't even come to play. On that interval, the "lower" branch of your "parabola" (y = -sqrt(2x + 6)) is the lower bound and the "upper" branch (y = sqrt(2x + 6)) is the upper bound. However when you go to the interval [-1, 5], y = x - 1 is is the lower limit and the "upper" branch of y^2 = 2x + 6 is the upper bound.