# Double integrals over general regions.

#### Petrus

##### Well-known member
Hello MHB,
Exemple 3: "Evaluate $$\displaystyle \int\int_D xy dA$$, where D is the region bounded by the line $$\displaystyle y=x-1$$ and parabola $$\displaystyle y^2=2x+6$$"
They say I region is more complicated (the x) so we choose y. so if we equal them we get $$\displaystyle x_1=-1$$ and $$\displaystyle x_2=5$$
then as they said it's more complicated if we work with x limit so we make them to y limit. and we got $$\displaystyle y=\sqrt{2x+6}$$ and $$\displaystyle y=x-1$$ the y limit shall be $$\displaystyle y_1=4$$ and $$\displaystyle y_2=-2$$ but if you put 5 in $$\displaystyle y=\sqrt{2x+6}$$ we get $$\displaystyle y=\pm 4$$ so how does this 'exactly' work. Shall I check the value on both function? I reread the chapter and try think and think but never come up with an answer why I cant use lower limit as $$\displaystyle -4$$
What I exactly mean why dont we have our lowest limit as -4 insted of -2
(It may be little confusing, sorry)

Regards,

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
but if you put 5 in $$\displaystyle y=\sqrt{2x+6}$$ we get $$\displaystyle y=\pm 4$$ so how does this 'exactly' work.
No , this is not correct .

#### Petrus

##### Well-known member
No , this is not correct .
Hello Zaid
$$\displaystyle y=\sqrt{16}$$ what do you mean

Regards,

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
$$\displaystyle \sqrt{16} \neq \pm 4$$

- - - Updated - - -

For this example it is better to separate the integral , we cannot define $$\displaystyle y=\sqrt{2x+6}$$ because $y$ takes negative values.

#### Petrus

##### Well-known member
$$\displaystyle \sqrt{16} \neq \pm 4$$

- - - Updated - - -

For this example it is better to separate the integral , we cannot define $$\displaystyle y=\sqrt{2x+6}$$ because $y$ takes negative values.
The point is we can get y value two way. $$\displaystyle y=x-1$$ and $$\displaystyle y=\sqrt{2x+6}$$ in the first one if we put 5 we get positive 4 so I can just put in first one ?