Should we consider negative axis when finding the volume?

[Phys12]

Question:
Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line. Sketch the
region, the solid, and a typical disk or washer.
y² = x; x = 2y; about the y-axis

Solution: http://www.slader.com/textbook/9780538497909-stewart-calculus-early-transcendentals-7th-edition/438/exercises/9/

Now, when I solved the question, I multiplies the entire integral by 2 since the limits I took were from -2 to 2. However, in the answer, they have considered only the values above the x-axis and not below. Why is it so? It doesn't specify in the question that we need to find the area of only that region which lies above the x-axis.

 

[Mark44]

Question:
Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line. Sketch the
region, the solid, and a typical disk or washer.
y² = x; x = 2y; about the y-axis

Solution: http://www.slader.com/textbook/9780538497909-stewart-calculus-early-transcendentals-7th-edition/438/exercises/9/

Now, when I solved the question, I multiplies the entire integral by 2 since the limits I took were from -2 to 2.

This doesn't make any sense. The two curves intersect at (0, 0) and (4, 2). What does your integral look like?

However, in the answer, they have considered only the values above the x-axis and not below. Why is it so? It doesn't specify in the question that we need to find the area of only that region which lies above the x-axis.

The region that is being revolved around the y-axis is completely above the x-axis, except the point (0, 0). Did you sketch the region that is being revolved?

 

[Phys12]

This doesn't make any sense. The two curves intersect at (0, 0) and (4, 2). What does your integral look like?

The region that is being revolved around the y-axis is completely above the x-axis, except the point (0, 0). Did you sketch the region that is being revolved?

Okay, I got it. Just one question, when we plot x=y^2, why don't we plot it below the x-axis as well? I know that it will cease to be a function, but we can values for x when we take y and -y, right? (That's what got me confused, I plotted y^2=x as y=x^2 but turned 90 degrees. Which made me draw another line below the x-axis as well for some reason-- stupid me!)

 

[Mark44]

Okay, I got it. Just one question, when we plot x=y^2, why don't we plot it below the x-axis as well?

You do plot a part of it below the x-axis. But, the region that is being revolved is between this curve and the line x = 2y. That region is completely in the first quadrant.

I know that it will cease to be a function, but we can values for x when we take y and -y, right? (That's what got me confused, I plotted y^2=x as y=x^2 but turned 90 degrees. Which made me draw another line below the x-axis as well for some reason-- stupid me!)