Math Problem Statement
Let R be the region bounded by y=x2 , x=0 , and y=4 . Set up the integral that represents the volume of the solid formed when R is revolved about the line x=4 using the Shell Method.
Solution
To set up the integral for the volume of the solid formed when the region is revolved about the line using the Shell Method, we proceed as follows:
Step 1: Analyze the given region
The region is bounded by:
- (a parabola),
- (the vertical line),
- (a horizontal line).
This means that the region starts from and extends to (since intersects at ).
Step 2: Shell Method formula
The formula for the volume of a solid using the Shell Method when revolving around a vertical line is:
In this case, we are revolving about .
-
Radius of each shell: The distance from a point to the line . This is given by . Since the region is between and , the radius simplifies to .
-
Height of each shell: The height is the difference between the top and bottom of the region at a given -value. The top of the region is given by , and the bottom of the region is given by . So, the height of the shell is .
Step 3: Set up the integral
Now, applying the Shell Method formula:
Final Answer:
Would you like to solve this integral step by step? Or do you have any questions about the setup?
Related Questions:
- How do you evaluate this integral?
- How does the Disk Method compare to the Shell Method for this problem?
- How do you apply the Shell Method when revolving around a horizontal line?
- What would the volume be if the region was revolved around the
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Volume of Solids
Shell Method
Formulas
V = 2π ∫(radius of shell)(height of shell) dx
Theorems
Shell Method
Suitable Grade Level
Grades 11-12
Related Recommendation
Volume of Revolution Using Cylindrical Shells: Rotating y = x^2 and y = 4 About the Y-Axis
Integral Setup for Volume Using Shell Method: Rotating y^2 Around x=9
Volume of Solid of Revolution using Shell Method - Rotating Region y = x about x = 9
Volume of Solid by Cylindrical Shells: Rotation Around x = 8
Volume of Solid of Revolution: Rotating y = 4x - x^2 Around x = 7