1. Volumes by Cylindrical Shells

2. What is the volume of and y=0 revolved around about the y-axis ? - since its revolving about the y-axis, the equation needs to be solved in terms of y - not easy to solve the equation in terms of y

3. Another Method Cylindrical shells can be used Cut into cylindrical sections and solve for the volume of the cylinder Volume = circumference x height x thickness

4. Radius is the x position Δr is Δx or dx The h is the y position or f(x)

6. Example Find the volume of the solid obtained by rotating the region bounded by and y=0 about the y-axis

7. Example Find the volume of the solid obtained by rotating the region bounded by y = x and y = x 2 about the y-axis