# Volume between two paraboloids on x and y axes.

• urnlint
In summary, the conversation discusses a calculus problem involving finding the volume between two paraboloids. The student is struggling to remember the proper steps and is looking for help. The tutor suggests using a hint involving x-y and x-y squared, and the student attempts to solve the problem by setting z=0 and finding the intersection in the xy plane. The student is unsure of the next steps and asks for help.
urnlint

## Homework Statement

Grr it got all got erased...This is a math problem that a student I tutor got wrong on his calculus test. I am having problems remembering what to do. I have looked up other problems, but the paraboloids seem simpler for some reason. Probably because someone is explaining it to me.

Find the volume between the two parabaloids x = $y^{2}$ + $z^{2}$ and y = $x^{2}$ + $z^{2}$

Hint: x - y divides $x^{2}$ - $y^{2}$ (I do not know how this helps. Probably because I am doing it wrong)

## The Attempt at a Solution

Set z = 0 to find the "shadow" in the xy plane: y = $x^{0.5}$ and y = $x^{2}$

They intersect at (0,0) and (1,1) because:
$x^{2}$ = $x^{0.5}$
$x^{2}$ - $x^{0.5}$ = 0
$x^{0.5}$($x^{1.5}$ - 1) = 0
$x^{0.5}$ = 0 gives x = 0 and y = 0 and
$x^{1.5}$ - 1 = 0
$x^{1.5}$ = 1 gives x = 1 and y =1

I think I understood up until now, but am sort of lost, and I do not have the answer given by the teacher to check myself. Do I do ∫∫∫ 1 dzdydx where 0≤x≤1, $x^{2}$≤y≤$x^{0.5}$ and for the interval of z I just solve one of the given paraboloids for z, which will give a ±square root? Like z = ±√(x - $y^{2}$) Then I would do trig substitution? Or would I subtract right from left so that I would just do a double integral of √(x - $y^{2}$) - √(y - $x^{2}$) with the above intervals for x and y?

I am sorry for the long post, but I am trying to remember what to do and this is the best way to explain it. Any help is greatly appreciated!

## What is the formula for finding the volume between two paraboloids on the x and y axes?

The formula for finding the volume between two paraboloids on the x and y axes is given by V = ∫∫∫ (f(x,y) - g(x,y)) dxdy, where f(x,y) and g(x,y) are the equations of the upper and lower paraboloids respectively.

## How do you determine the boundaries for the triple integral when calculating the volume between two paraboloids on the x and y axes?

The boundaries for the triple integral can be determined by finding the intersection points of the two paraboloids. These points will serve as the limits for the x and y axes in the integral.

## What is the relationship between the volume between two paraboloids on the x and y axes and the volume of a solid with circular cross sections?

The volume between two paraboloids on the x and y axes is equal to the volume of a solid with circular cross sections when the cross sections are taken perpendicular to the x or y axis. This is because the cross sections will be circular in shape for both the paraboloids and the solid.

## Can the volume between two paraboloids on the x and y axes be negative?

No, the volume between two paraboloids on the x and y axes cannot be negative. Volume is a measure of the amount of space occupied by an object, and it cannot have a negative value.

## How do you interpret the volume between two paraboloids on the x and y axes in terms of the physical world?

The volume between two paraboloids on the x and y axes can represent the space between two curved surfaces in the physical world, such as the space between two hills or two curved roofs. It can also be used to calculate the volume of a container or a tank with a curved bottom.

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