Augustin Louis Cauchy 1789 – 1857 Augustin Louis Cauchy 1789 – 1857 Cauchy pioneered the study of analysis, both real and complex, and the theory of permutation.

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Augustin Louis Cauchy 1789 – 1857 Augustin Louis Cauchy 1789 – 1857 Cauchy pioneered the study of analysis, both real and complex, and the theory of permutation groups. He also researched in convergence and divergence of infinite series, differential equations, determinants, probability and mathematical physics.

x x y R x

x x A(x)A(x)

x A(x)A(x)

Find the volume of the solid below with a circular base. For this solid, each cross section made with a plane perpendicular to the x-axis is an equilateral triangle.

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 disks (flat cylinders) and add their volumes. The volume of each flat cylinder (disk) is the area the face times thickness: In this case: r = the y value of the function thickness = a small change in x = dx x

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

In summary, if the bounded region is rotated about the x-axis, and the plane perpendicular to the x-axis intersects the region with circular disks, then the formula is: A shape rotated about the y-axis would be: Area of Face Thickness. Volume of the Disk

The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. 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.

x

Area of Face

x

a) b) What is the volume of the solid that results when R is revolved about the line y = 3.

The region bounded by and is revolved about the line x = 2. Find the volume.

The outer radius is: rOrO The inner radius is: riri The region bounded by and is revolved about the line x = 2. Find the volume. x = 2

outer radius inner radius thickness of slice cylinder Find the volume of the region bounded by,, and revolved about the y -axis.

a) What is the volume of the solid that results when R is revolved about the line x-axis. b) What is the volume of the solid that results when R is revolved about the line y =  2. c)