Math Problem Statement
Do question 27
Solution
Question 27 asks to "find the volume of a cone of height and base radius by integration." Let me explain how to solve this.
Solution:
The volume of the cone can be derived using the concept of integration. A cone is a solid of revolution generated by rotating a line segment (forming the side of the cone) around the -axis.
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Equation of the Line:
If the cone has height and radius , we can place the cone in a coordinate system where:- The vertex is at ,
- The base is at ,
- The slope of the line defining the side of the cone is .
Thus, the equation of the line is:
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Volume Formula for Solids of Revolution:
The volume of a solid of revolution about the -axis is given by: -
Substitute :
Substitute into the formula: -
Simplify the Integral:
Expand and simplify:The integral of is:
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Evaluate the Definite Integral:
Evaluate from to : -
Simplify the Expression:
Cancel and simplify:
Final Answer:
The volume of the cone is:
Would you like further clarification or related problems? Here are some questions to explore further:
- How do you find the volume of a sphere using integration?
- Can you find the surface area of a cone using integration?
- How is the formula for the volume of a cone different from that of a cylinder?
- What happens to the volume of the cone if the height is doubled?
- Can you calculate the volume for a cone with specific values of and ?
Tip: Always sketch the problem to understand the geometry better before setting up the integral!
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Volumes of Revolution
Formulas
Volume of a solid of revolution: V = π∫[y(x)]² dx
Equation of a line: y = mx + b
Integral of x^n: ∫x^n dx = (x^(n+1))/(n+1)
Theorems
Fundamental Theorem of Calculus
Properties of Definite Integrals
Suitable Grade Level
Grades 11-12, Undergraduate