Find Volume of Solid: Integral Rotation | y=1+sec x & y=3

In summary, the problem asks to find the volume of the solid formed by rotating the region bounded by the curves y=1+sec(x) and y=3 around the line y=1. The region is an arch of the graph of y=1+sec(x) up to the line y=3. The problem is not clearly defined and more information is needed to solve it accurately.
  • #1
iRaid
559
8

Homework Statement


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.

[tex]y=1+sec x[/tex] [tex]y=3[/tex] about [tex]y=1[/tex]

Homework Equations


The Attempt at a Solution


I don't understand how to do this since y=3 crosses at infinite points. I know that is crosses at -∏/3 and ∏/3.
 
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  • #2
iRaid said:

Homework Statement


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.

[tex]y=1+sec x[/tex] [tex]y=3[/tex] about [tex]y=1[/tex]

Homework Equations


The Attempt at a Solution


I don't understand how to do this since y=3 crosses at infinite points. I know that is crosses at -∏/3 and ∏/3.

My guess is that the region to be rotated around the line y = 1 is just a single arch of the graph of y = 1 + sec(x), up to the line y = 3. The problem should have been more specific in describing the region, IMO.
 

1. How do you find the volume of a solid using integral rotation?

To find the volume of a solid using integral rotation, you must first identify the function that represents the shape of the solid. Then, use the formula V = π∫a^2-b^2 dx to calculate the volume, where a and b are the limits of integration and dx represents the thickness of the slices.

2. What is the difference between shell and disk method for finding volume?

The shell method and disk method are two different techniques used to find the volume of a solid using integral rotation. The shell method involves integrating along the axis of rotation, while the disk method involves integrating perpendicular to the axis of rotation. The result is the same, but the approach and setup may differ.

3. How do you determine the limits of integration for a solid with a curved boundary?

To determine the limits of integration for a solid with a curved boundary, you must first plot the function and identify the points of intersection with the axis of rotation. These points will serve as the limits of integration. You may need to use trigonometric identities or calculus techniques to solve for these points.

4. Can you use integral rotation to find the volume of a solid with a non-circular cross section?

Yes, integral rotation can be used to find the volume of a solid with a non-circular cross section. The formula for calculating volume remains the same, but the limits of integration may be more complex and may require the use of multiple integrals.

5. What is the significance of the thickness of the slices in integral rotation?

The thickness of the slices in integral rotation represents the precision of the calculation. The thinner the slices, the more accurate the result will be. However, using extremely thin slices may make the calculation more complex and time-consuming. It is important to balance accuracy and efficiency when choosing the thickness of slices.

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