Volume of Solid: Find Y-Axis Rot. Region

In summary, the conversation discusses finding the volume of a solid formed by rotating a given region around the y-axis. The problem has been approached by breaking it into two parts - solving for a cylinder (or shell) from y=0 to 5 and solving for a washer from y=5 to e^(3/2)+5. The answer provided by the person is (1.25pi) for the cylinder or disk and (.49811372pi) for the washer. The conversation also clarifies that the cylinder elements are parallel to the y-axis and the natural variable for that is x.
  • #1
africanmasks
12
0

Homework Statement



Find the volume of the solid formed by rotating the region enclosed by the following equations about the Y-AXIS.

y= e^(3x)+5
y=0
x=0
x= 1/2

Homework Equations


The Attempt at a Solution



I keep getting the answer wrong. I broke the problem into two parts: solved a cylinder(disk) from y= 0 to 5 and solved a washer from y=5 to e^(3/2)+5

My answer was (1.25pi) (for cylinder or disk) + (.49811372pi) (for washer)
 
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  • #2
Where's your work? For the cylinder wouldn't x go from 0 to 1/2?
 
  • #3
you're rotating around the y not x
 
  • #4
africanmasks said:
you're rotating around the y not x

Yes, so the cylinder elements are parallel to the y axis, sometimes called "dx elements". Your natural variable for that is x.

Maybe I misunderstand your terminology. Is what you call a cylinder what some texts call a shell?
 
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FAQ: Volume of Solid: Find Y-Axis Rot. Region

1. What is the formula for finding the volume of a solid in a y-axis rotation region?

The formula for finding the volume of a solid in a y-axis rotation region is ∫(π * r^2)dx, where r is the distance from the y-axis to the outer edge of the solid and dx is the width of each infinitesimally thin slice of the solid.

2. How do you determine the limits of integration for finding the volume in a y-axis rotation region?

The limits of integration for finding the volume in a y-axis rotation region are determined by the points at which the solid intersects with the y-axis. These points can be found by setting the equation of the solid equal to zero and solving for x.

3. Can you use the same formula to find the volume of a solid in a different axis rotation region?

No, the formula for finding the volume of a solid in a y-axis rotation region is specific to that particular region. Different formulas are used for finding the volume in other axis rotation regions, such as the x-axis or z-axis.

4. How does the shape of the solid affect the volume in a y-axis rotation region?

The shape of the solid greatly affects the volume in a y-axis rotation region. A solid with a larger radius will have a greater volume, while a solid with a smaller radius will have a smaller volume.

5. Can you use calculus to find the volume of a solid in a y-axis rotation region?

Yes, calculus is used to find the volume of a solid in a y-axis rotation region by using integration to sum up the infinitesimally thin slices of the solid and find the total volume.

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