Volume of the region between two parabolas?

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In summary, the task is to find the volume of the region enclosed by the parabolas z = 1 - y^2 and z = y^2 - 1 for x between 0 and 2. The y bounds are from -1 to 1 where the parabolas meet, and the x bounds are from 0 to 2. The integral can be expressed as a double integral with x from 0 to 2 and y from -1 to 1, of the function (y^2 - 1) - (1 - y^2). The x coordinate can also be expressed as f(y,z)=2 and then integrated with respect to y and z within the specified bounds.
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Homework Statement



Find the volume of the region enclosed by z = 1 - y^2 and z = y^2 - 1 for x lying between 0 and 2 inclusive.

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The Attempt at a Solution



I know that the y bounds are from -1 to 1, where the parabolas meet. x bounds are from 0 to 2. So would the integral simply by the double integral , x from 0 to 2, y from -1 to 1 of the (y^2 -1) - (1 - y^2)? I'm confused at this point. All the limits of integration are constant..
 
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Ah no worries, found out how to solve it! I expressed the x coordinate as f(y,z)=2. Then integrated this function with respect to y and z, with the bounds I specified. Thanks!
 

FAQ: Volume of the region between two parabolas?

What is the formula for finding the volume of the region between two parabolas?

The formula for finding the volume of the region between two parabolas is ∫(upper curve - lower curve) dx. This means taking the integral of the difference between the equations of the two parabolas with respect to x.

How do you determine the upper and lower curves when finding the volume between two parabolas?

When finding the volume between two parabolas, the upper curve is typically the parabola with the larger coefficient of x^2, and the lower curve is the one with the smaller coefficient of x^2. However, it is important to graph the two parabolas to confirm which is the upper and lower curve.

Can the volume between two parabolas be negative?

Yes, the volume between two parabolas can be negative. This occurs when the upper curve is below the lower curve in certain regions, resulting in a negative value when the integral is evaluated.

Is there a specific method for finding the volume between two parabolas?

Yes, there is a specific method for finding the volume between two parabolas. This method involves taking the integral of the difference between the two parabolas and evaluating it within the bounds of the region of interest. It is also important to correctly identify the upper and lower curves, as well as any points of intersection between the two parabolas.

Can the volume between two parabolas be calculated using other methods besides integration?

Yes, there are other methods that can be used to calculate the volume between two parabolas, such as using the disk or washer method from calculus. However, integration is the most common and efficient method for finding the volume in this scenario.

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