Cylindrical shells offer a powerful method for calculating volumes of revolution. By slicing solids into thin vertical rectangles and rotating them, we can integrate to find the total volume. This approach is especially useful for regions bounded by functions of x rotated around vertical axes.
Choosing between cylindrical shells and disk/washer methods depends on the problem setup. Cylindrical shells excel with vertical rotations and multiple x-functions, while disk/washer shines for horizontal rotations. Visualizing the solid helps determine the best approach for each unique situation.
Volumes of Revolution: Cylindrical Shells
Cylindrical shells volume calculation
Top images from around the web for Cylindrical shells volume calculation
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
1 of 3
Top images from around the web for Cylindrical shells volume calculation
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
Volumes of Revolution: Cylindrical Shells · Calculus View original
Is this image relevant?
1 of 3
Calculates volume of solid generated by rotating region bounded by curves around vertical axis
Divides region into thin vertical rectangles forming cylindrical shells when rotated
Approximates volume of each shell using formula 2πrhΔx
r represents distance from axis of rotation to shell's center
h represents height of shell determined by function f(x)
Δx represents width of shell
Calculates total volume by integrating volumes of all shells using V=∫ab2πrhdx
a and b represent x-coordinates of region's boundaries (interval of integration)
Determines r and h based on axis of rotation and bounding function(s)
For rotation around y-axis, r=x and h=f(x)
For rotation around vertical line x=a, r=∣x−a∣ and h=f(x)
Examples:
Rotating region bounded by y=x2 and y=4 around y-axis
Rotating region bounded by y=sin(x) and x-axis from x=0 to x=π around line x=π
Method selection for revolution volumes
Chooses cylindrical shell method for regions bounded by functions of x rotated around vertical axis
Particularly useful for regions bounded by multiple functions of x (y=x2 and y=x3)
Prefers disk/washer method for regions bounded by functions of y rotated around horizontal axis
Also suitable for regions bounded by single function of x rotated around horizontal axis (y=x rotated around x-axis)
Considers complexity of integral when selecting method
Aims to simplify integral by choosing appropriate method based on given functions and axis of rotation
Sketches region and solid of revolution to visualize problem and determine most suitable method
Rotating region bounded by y=ex and y=0 from x=0 to x=1 around y-axis (cylindrical shell method)
Rotating region bounded by y=x2 and x=1 around x-axis (disk/washer method)
Off-axis rotation volume calculation
Adjusts formula for volume of each shell when rotating around vertical line x=a (other than y-axis)
Calculates radius r as distance between shell's center and line x=a using r=∣x−a∣
Maintains height h determined by function f(x)
Determines x-coordinates of region's boundaries (a and b) based on given functions
Integrates volumes of all shells using modified formula V=∫ab2π∣x−a∣f(x)dx
Evaluates integral to find total volume of solid of revolution
Uses absolute value when calculating radius to ensure positive volume
Sketches region and solid of revolution to understand problem and necessary formula adjustments
Examples:
Rotating region bounded by y=x2 and y=4 around line x=2
Rotating region bounded by y=cos(x) and x-axis from x=0 to x=π/2 around line x=π
Fundamentals of Volumes of Revolution
Volume represents the three-dimensional space occupied by a solid of revolution
Solid of revolution is created by rotating a two-dimensional region around an axis
Cross-section of a solid of revolution is the shape formed when cutting perpendicular to the axis of rotation
Definite integral is used to sum up infinitesimal volumes to calculate the total volume
Area of the region being rotated directly influences the volume of the resulting solid