Should we consider negative axis when finding the volume?

In summary, the question asks for the volume of a solid obtained by rotating the region bounded by the curves y^2 = x and x = 2y about the y-axis. The solution provided considers only the values above the x-axis because the region being revolved is completely above the x-axis, except for the point (0,0). The region is in the first quadrant and the confusion may have arisen from plotting the curve y^2 = x as y = x^2 instead of y = sqrt(x).
  • #1
Phys12
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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.
 
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  • #2
Phys12 said:
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?
Phys12 said:
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?
 
  • #3
Mark44 said:
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!)
 
  • #4
Phys12 said:
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.
Phys12 said:
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!)
 
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1. What is a negative axis and why should we consider it when finding volume?

A negative axis is a line or plane on a graph that extends in the negative direction (left or down). We should consider it when finding volume because it can result in negative values, which may not make sense in the context of the problem we are trying to solve.

2. How does considering negative axis affect the calculation of volume?

When considering negative axis, we must take into account any negative values that may result from the calculation of volume. This may require us to adjust our formula or approach in order to accurately represent the physical space or object we are measuring.

3. Can we ignore the negative axis when finding volume?

No, we cannot ignore the negative axis when finding volume. Neglecting to consider the negative axis can lead to incorrect or nonsensical results in our calculation, which can significantly impact the accuracy of our findings.

4. Are there any instances where negative axis should not be considered when finding volume?

Yes, there may be instances where negative axis does not need to be considered when finding volume, such as when we are dealing with a one-dimensional object that does not extend in the negative direction. However, it is always important to carefully consider the context of the problem and determine if negative axis should be taken into account.

5. How can we represent negative axis in our calculations?

There are a few ways we can represent negative axis in our calculations, such as using negative signs (-) or explicitly stating the negative direction in our equations. It is important to be consistent and clearly communicate the representation of negative axis to avoid confusion or errors in our calculations.

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