Mistake in a calculus technique? I'm making a revision of calculus using Stewart's book and I'm studying right now some classical integral applications techniques:

I think if we do this calculation using $x$ as a variable in the integral we should have a different result, see:

The first drawing is the area when we calculate the integral in $x$ and the second when we calculate the integral in $y$.
Note that the $y$ part above the $x$ axis is positive and below negative while $x$ is positive in the right part and negative in the left part.
So we can't have the same result in both integrals because the areas should be different.
Thanks
 A: I don't quite understand what you mean by things being different. 
On your picture in the book, if you choose to integrate with respect to $x$, you would integrate the top function minus the bottom function. When you do this, it doesn't matter if the graph of the function(s) is(are) below the $x$-axis or not. Since you are taking top minus bottom, the difference will always be the positive distance. And you want to integrate a positive distance to get the (positive) area.
When you integrate with respect to $y$ you, likewise, want to integrate right minus left. Why? Because, again, you want the difference to be positive.
In you example, as the book notes, integrating with respect to $x$ is requires more work since you would have to split up the integer $[-3,5]$ into two: $[-3,-1]$ and $[-1,5]$ on each of these intervals you need to find the equation of the top and the bottom. So, for example, for the first interval, the top would be given by $\sqrt{2x +6}$ adn the bottom would be $-\sqrt{2x+6}$. So the integral over the first interval is
$$
\int_{-3}^{-1} \sqrt{2x+6} - (-\sqrt{2x+6})\; dx.
$$
And then you have the second integral From $-1$ to $5$:
$$
\int_{-1}^5 \sqrt{2x+6} - (x-1)\; dx.
$$
So you have to split up the integral. 
By integrating with respect to $y$ you don't have to do this (as illustrated in the book).
In all cases, you get the same result. And both methods "don't care" about whether or not the functions are positive or negative. 
