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7.3 Day One: Volumes by Slicing. 3. 3. 3. 0. h. 3. Find the volume of the pyramid:. Consider a horizontal slice through the pyramid. 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 .
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3 3 3 0 h 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. The volume of the slice is s2dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. This correlates with the formula: s dh
1 Method of Slicing: Sketch the solid and a typical cross section. Find a formula for V(x). (Note that I used V(x) instead of A(x).) 2 3 Find the limits of integration. 4 Integrate V(x) to find volume.
y x If we let h equal the height of the slice then the volume of the slice is: h 45o x A 45o 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. Since the wedge is cut at a 45o angle: Since
y x Even though we started with a cylinder, p 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 p
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.
The volume of each flat cylinder (disk) is: 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. 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 The radius is the x value of the function . We use a horizontal disk. The thickness is dy. 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 volume of the washer is: The region bounded by and is revolved about the y-axis. Find the volume. If we use a horizontal slice: The “disk” now has a hole in it, making it a “washer”. outer radius inner radius
The washer method formula is: 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.
r R If the same region is rotated about the line x=2: The outer radius is: The inner radius is: p
Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cosx 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.
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: cylinder inner radius outer radius thickness of slice Japanese Spider Crab Georgia Aquarium, Atlanta
Here is another way we could approach this problem: cross section If we take a vertical slice and revolve it about the y-axis we get a cylinder. If we add all of the cylinders together, we can reconstruct the original object.
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.
This is called the shell method because we use cylindrical shells. cross section If we add all the cylinders from the smallest to the largest:
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.
If we take a vertical slice and revolve it about the y-axis we get a cylinder. Shell method:
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. p
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 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 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 = x3 + x + 1, y = 1, and x = 1 about the line x = 2.
Find the volume of the solid formed by revolving the region bounded by the graphs y = x3 + x + 1, y = 1, and x = 1 about the line x = 2.