3 3 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. s dh The volume of the slice is s 2 dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. 0 3 h This correlates with the formula:

Method of Slicing: 1 Find a formula for A ( x ). Sketch the solid and a typical cross section. 2 3 Find the limits of integration. 4 Integrate V ( x ) to find volume.

Volume formula :: Let S be a solid bounded by two parallel planes perpendicular to the x-axis at x=a and x=b. If, for each x in [a,b], the cross-sectional area of S perpendicular to the x-axis is A(x), then the volume of the solid is provided A(x) is integrable.

Volume formula :: Let S be a solid bounded by two parallel planes perpendicular to the y-axis at y=c and y=d. If, for each y in [c,d], the cross-sectional area of S perpendicular to the y-axis is A(y), then the volume of the solid is provided A(y) is integrable.

x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge shape several ways, but the simplest cross section is a rectangle. If we let h equal the height of the slice then the volume of the slice is: Since the wedge is cut at a 45 o angle: x h 45 o Since

x y Even though we started with a cylinder,  does not enter the calculation!

Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections 

Suppose we start with this curve. If we want to build a nose cone in this shape. So we put a piece of wood in a lathe and turn it to a shape to match the curve.

How could we find the volume of the cone? One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes. The volume of each flat cylinder (disk) is: In this case: r= the y value of the function thickness = a small change in x = dx

The volume of each flat cylinder (disk) is: If we add the volumes, we get:

This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk. If the shape is rotated about the x-axis, then the formula is: A shape rotated about the y-axis would be:

Find the volume of the solid generated by revolving the regions about the x-axis. bounded by

Find the volume of the solid generated by revolving the regions about the x-axis.bounded by

Find the volume of the solid generated by revolving the regions about the y-axis. bounded by

Find the volume of the solid generated by revolving the regions about the x-axis.bounded by

Find the volume of the solid generated by revolving the regions about the line y = -1.bounded by

The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

The region bounded by and is revolved about the y-axis. Find the volume. The “disk” now has a hole in it, making it a “washer”. If we use a horizontal slice: The volume of the washer is: outer radius inner radius

This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. The washer method formula is:

If the same region is rotated about the line x = 2 : The outer radius is: R The inner radius is: r 