7.3 Day One: Volumes by Slicing

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 V ( x ). (Note that I used V ( x ) instead of A(x).) Sketch the solid and a typical cross section. 2 3 Find the limits of integration. 4 Integrate V ( x ) to find volume.

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 

Cavalieri’s Theorem: Volume of a SphereVolume of a Sphere 

7.3 Disk and Washer Methods

Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. So I 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:

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 

Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

7.3 The Shell Method

Find the volume of the region bounded by,, and revolved about the y - axis. We can use the washer method if we split it into two parts: outer radius inner radius thickness of slice cylinder Japanese Spider Crab Georgia Aquarium, Atlanta

If we take a vertical sliceand revolve it about the y-axis we get a cylinder. cross section If we add all of the cylinders together, we can reconstruct the original object. Here is another way we could approach this problem:

cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.

cross section If we add all the cylinders from the smallest to the largest: This is called the shell method because we use cylindrical shells.

Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

Shell method: If we take a vertical slice and revolve it about the y-axis we get a cylinder.

Note:When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method. 

Find the volume of the solid when the region bounded by the curve y =, the x-axis, and the line x = 4 is revolved about the x-axis. Find the volume of the solid using cylindrical shells.

Find the volume of the solid of revolution formed by revolving the region bounded by the graph of and the y axis, 0 ≤ y ≤ 1, about the x-axis. Use the Shell Method.

Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.