Section 6.3

Ex #4 pg. 396

Find the volume of the solid obtained by rotating the region bounded by and

y = 0 about the line x = 2.

(1) Draw, Label, Points of Intersection , if x = 0 and x = 1

(2) Cylindrical Shell Method (Why is this best option?)

(Begin w/Rectangle parallel to the axis of revolution.)

Requires width of the rectangle: OR

Average radius of the shell (part of ):

(Distance from center of rectangle to the axis of revolution)

=

Height of the rectangle: =

(3) Set Up the Integral and Evaluate:

= ... =

Ex (HW #31)

The given interval represents the volume of a solid. Describe the solid

Shell : Hint: 4 parts w/in the integral the first of which is which indicates the formula .

r(y) placed second, r(y) = 3-y. This is the distance from the axis of revolution to the center of the rectangle/shell. Axis of revolution would be y = 3. [Note: If the axis of revolution was y = -3, r(y) would be y+3.

h(y) is the height of the rectangle/shell which is placed third. h(y) is the function or the difference of two functions. Simplest case is which is a sideways parabola, another option is using two functions and x = 1.

Summary:

The solid is obtained by rotating the region bounded by

(i) , x = 0 and y = 0 about the line y = 3 using cylindrical shells.

OR

(ii) , x = 1 and y = 0 about the line y = 3 using cylindrical shells.