1. Volumes

2. Right Cylinder

4. Volume of a Right Cylinder (Slices) Cross section is a right circular cylinder with volume Also obtained as a solid of revolution Consider the case when h=5 and r=2

5. Volume of a Rectangular Box Cross section is a right cylinder with volume Consider the case of a box with dimensions 3x5x7 Area of a cross section is b*l

6. Volume of a Sphere of radius r (slices) Cross section is a right cylinder with volume Consider the case of a sphere of radius r=3

7. Another Example Sketch the solid of revolution obtained by rotating around x-axis and find its volume Sketch the solid of revolution obtained by rotating around y-axis and find its volume Sketch the solid of revolution obtained by rotating around x=5 and find its volume

8. Volume using Shells

9. Example Sketch the solid of revolution obtained by rotating about x-axis and find its volume Sketch the solid of revolution obtained by rotating around y-axis and find its volume Sketch the solid of revolution obtained by rotating around x=1.2 and find its volume

10. Cylindrical Shells Definition of a shell Animations

11. Formulas Shell in general position Volume of shell

12. Volume Formulas Function y=f(x) Rotating about y-axis Function x=f(y) Rotating about x-axis