Volume of revolution, region bounded by two functions

In summary, the volume of revolution is the volume of a three-dimensional solid formed by rotating a two-dimensional region about an axis. It can be calculated using the method of cylindrical shells or the method of disks/washers. A region bounded by two functions is a two-dimensional region enclosed by two curves represented by two functions. The boundaries of this region can be found by setting the two functions equal to each other or visually on a graph. The concept of volume of revolution has many real-world applications in fields such as engineering, architecture, and physics.
  • #1
Kqwert
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Homework Statement


Let R be the area in the xy-plane in the 1st quadrant which is bounded by the curves y^2+x^2 = 5, y = 2x and x = 0. (y-axis). Let T be the volume of revolution that appears when R is rotated around the Y axis. Find the volume of T.

Homework Equations

The Attempt at a Solution


This is the integral I have put up, using the cylindrical shell method:

int(x*(sqrt(5-x^2)-2x)*2pi)dx from 0 to 1. Is this the correct integral?
 
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  • #2
Looks right to me.
 

FAQ: Volume of revolution, region bounded by two functions

What is the volume of revolution?

The volume of revolution refers to the volume of a three-dimensional solid formed by rotating a two-dimensional region about an axis.

How is the volume of revolution calculated?

The volume of revolution can be calculated using the method of cylindrical shells, which involves integrating the cross-sectional area of the solid along the axis of rotation. Alternatively, it can also be calculated using the method of disks or washers, which involves integrating the area of the cross-section perpendicular to the axis of rotation.

What is a region bounded by two functions?

A region bounded by two functions is a two-dimensional region on a coordinate plane that is enclosed by two curves. These two curves can be represented by two functions, one for the upper boundary and one for the lower boundary.

How do you find the boundaries of a region bounded by two functions?

To find the boundaries of a region bounded by two functions, you can set the two functions equal to each other and solve for the x-values where they intersect. These x-values will be the boundaries of the region on the x-axis. Additionally, you can also look at the graphs of the two functions to determine the boundaries visually.

Are there any real-world applications of volume of revolution?

Yes, the concept of volume of revolution has many real-world applications, such as in engineering, architecture, and physics. For example, it can be used to calculate the volume of a water tank or the volume of a curved pipe. In physics, it is used to calculate the moment of inertia for rotating objects.

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