Math Problem Statement
47–61 Find the volume of the described solid S. 47. A right circular cone with height h and base radius r using integration
Solution
To find the volume of a right circular cone using integration, we can slice the cone horizontally and sum up the volumes of thin disks with infinitesimal thickness.
Steps to Solve:
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Equation of the Cone:
- A cone has a base radius and a height .
- Let's position the cone so that its vertex is at the origin and its height extends along the -axis up to . The radius at the base (where ) is .
- At a height , the radius of the cross-section is proportional to . From similar triangles, the radius at any height is:
-
Volume of a Thin Disk:
- Consider a thin slice of the cone at height with thickness . The radius of this disk is .
- The volume of this thin disk is:
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Integrating to Find Total Volume:
- To get the total volume of the cone, integrate the volume of the disks from to :
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Performing the Integration:
- The integral is:
- The integral of is:
- Substituting this back:
- The integral is:
Thus, the volume of the cone is:
This is the standard formula for the volume of a cone.
Let me know if you'd like further details or have any questions!
5 Related Questions:
- How would the method change if the cone was an oblique cone instead of a right cone?
- What would be the volume of a cone with a slant height instead of vertical height?
- How can you extend this integration approach to find the volume of a cone with an elliptical base?
- Can you derive the volume of a cylinder using a similar method of integration?
- How can you compute the surface area of this cone using integration?
Tip:
When solving problems involving solids of revolution, setting up the geometry carefully with correct limits and functions is crucial for successful integration.
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Solid Geometry
Formulas
V = (1/3)πr²h (Volume of a cone)
R(y) = (r/h)y (Radius of cross-section at height y)
Theorems
Volume of Solids of Revolution
Suitable Grade Level
Grades 11-12 (High School Calculus)