1. Drill: Find the area in the 4 th quadrant bounded by y=e x -5.6; Calculator is Allowed! 1) Sketch 2) Highlight 3) X Values 4) Integrate X=? X=0 X=1.723

2. How do we find the area between curves? March 4, 2015 Do Now: Previous Slide Agenda: 1. Do Now 2. HW Questions 3. Volume and Calculus the methods 4. Slicing method notes 5. Homework Take out: pencil, hw, notebook Homework: 406 1, 2 a and b only, 3 Objectives: You will be able to find the volume of solid figure using cross sections.

3. A Real Life Situation Damn, that’s a lotta toilet paper! I wonder how much is actually on that roll? Relief

4. How do we get the answer? CALCULUS!!!!! (More specifically: Volumes by Integrals)

5. Volume and Calculus We are going to look at 3 main ways to solve for the volume of a rotating figure. 1.Volume of Cross sections (slicing method /Disk method(curves). 2.Washer Method (AKA. the dougnut method) 3.Shell Method (cylindrical shells)

6. When do we use which method? Slicing method –no axis of rotation/revolution Disk method- If the axis of revolution is the boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution. (basically you only have one function) Washers method- If the axis of revolution is not a boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution. (basically you have 2 or more curves)Washers- easier when rotating about x-axis or when rotating about lines y=… (horizontal lines). Shell method- If the cross sections of the solid are taken parallel to the axis of revolution See above. When rotating about the y-axis or when rotating x=… (vertical lines). NOTE: Shell and washer are interchangeable can use for rotational solids. But depending on functions and the capability of isolating x and y.

7. Volume by Slicing Volume = length x width x height Total volume of loaf of bread=  (A x  t) Volume of a slice = Area of a slice x Thickness of a slice A tt

8. Volume by Slicing Total volume =  (A x  t) VOLUME =  A dt But as we let the slices get infinitely thin, Volume = lim  (A x  t)  t  0 Recall: A = area of a slice

9. Volume and Calculus n Volume of a Solid The volume of a solid of known integrable cross section area A(x) from x = a to x = b is the integral of A from a to b,

10. Volumes n How to Find Volume by the Method of Slicing 1. Sketch the solid and a typical cross section. 2. Find a formula for A(x). 3. Find the limits of integration. 4. Integrate A(x) to find the volume.

11. Example Square Cross Sections

13. Volume by perpendicular square cross sections.

15. The solid lies between planes perpendicular to the x-axis. The cross sections are semicircular disks with bases in the xy-plane. Find the volume using cross sections.

16. Find the volume of a solid whose base is the circle x 2 + y 2 = 4 and where cross sections perpendicular to the x-axis are all equilateral triangles whose sides lie on the base of the circle.

17. Find the volume of a solid whose base is the circle x 2 + y 2 = 4 and where cross sections perpendicular to the x-axis are all Isosceles right triangles whose sides lie on the base of the circle.

18. You try one more (Calculator OK)

19. Answer Integrate on your calculator