# Volume of solid rotated around y=1

• jaguar ride
In summary, the conversation discusses finding the volume of a solid formed by revolving a given region around a line. The equations and attempts at a solution are provided, and it is suggested to sketch the region to identify incorrect integration limits. After realizing the limits could be negative, the correct solution is found to be 16∏/15.
jaguar ride

## Homework Statement

Find the volume of the solid formed by revolving the region bounded by f(x) = 2-x^2 and g(x) = 1 about the line y = 1.

## Homework Equations

V = ∏∫(1-f(x))^2dx - ∏∫(1-g(x))^2dx

## The Attempt at a Solution

I keep ending up with ∏∫(1-(2-x^2))^2dx - ∏∫(1-1)^2dx, on the interval from x = 0, x = 1. This gives me ∏∫[(-1+x^2)^2 - 0]dx
= ∏[x - (2/3)x^3 + (1/5)x^5].

This keeps giving me 8∏/15.
This is on a test review sheet, with the answer being 16∏/15. So I can get that answer by multiplying mine by 2, but why would I do that? I must be doing something wrong somewhere else. Any help would be greatly appreciated! Thanks.

Hi jaguar ride! Welcome to PF!

jaguar ride said:
but why would I do that? I must be doing something wrong somewhere else.

Sketch a plot of the given region. ;)

You will notice that the integration limits are incorrect.

Ah yes. Should have mentioned that a sketch is the first thing I do when solving these problems.
That being said, I feel extra stupid for staring at this thing for so long and not realizing the limits could be negative.

Got the answer. Makes sense that I could multiply it by 2 now...

Thanks for your help and the warm welcome, I'll be back for physics help soon enough...

jaguar ride said:
Ah yes. Should have mentioned that a sketch is the first thing I do when solving these problems.
That being said, I feel extra stupid for staring at this thing for so long and not realizing the limits could be negative.

Got the answer. Makes sense that I could multiply it by 2 now...

Thanks for your help and the warm welcome, I'll be back for physics help soon enough...

## 1. What is the volume of a solid rotated around y=1?

The volume of a solid rotated around y=1 can be calculated using the formula V=π∫(y-1)^2 dx, where the limits of integration are the x-values of the intersection points between the solid and the line y=1.

## 2. How do you determine the intersection points between a solid and the line y=1?

To determine the intersection points, set the equation of the solid equal to y=1 and solve for the x-values. These x-values will be the limits of integration in the volume formula.

## 3. Can the volume of a solid rotated around y=1 be negative?

No, the volume of a solid rotated around y=1 cannot be negative. The volume of a solid is always a positive value, representing the amount of space the solid occupies.

## 4. What happens if the solid intersects with the line y=1 multiple times?

If the solid intersects with the line y=1 multiple times, the volume formula needs to be split into separate integrals for each section. The limits of integration will be the x-values of each intersection point.

## 5. Can you use the same formula to calculate the volume of a solid rotated around a different line?

No, the formula V=π∫(y-1)^2 dx is specific to solids rotated around the line y=1. To calculate the volume of a solid rotated around a different line, a different formula would need to be used.

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