Volumes by Cylindrical Shells2

The radius of a cylindrical shell is the distance from the axis of rotation to the shell itself, often represented as a function of x, such as $$r = 3 - x$$ .

The height of a cylindrical shell is determined by the difference between the upper and lower functions that define the region being revolved.

The volume of a cylindrical shell is given by the formula: $$V = 2\pi \int_{a}^{b} (radius)(height) \, dx$$ .

The integral limits represent the bounds of the region being revolved around the axis, typically the x-values where the region starts and ends.

The radius can be expressed as a function of x, such as $$r = 3 - x$$ , indicating how the radius decreases or increases as x changes.

The axis of rotation is the line around which the region is revolved to create the cylindrical shell, affecting the radius and volume calculations.

To set up an integral for volume using cylindrical shells, identify the radius and height functions, then integrate the product of these functions multiplied by 2π over the interval.

The disk method slices the solid perpendicular to the axis of rotation, while the shell method slices parallel to the axis, using cylindrical shells.

An example function could be $$y = 4 - x^2$$ , which defines the upper boundary of the region being revolved.

The function defines the height of the cylindrical shell, which is the vertical distance between the upper and lower curves.