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 LIMERICK GENERATING STATION Limerick Generating Station, located in Limerick Township, Montgomery County, PA, is a two-unit nuclear generation facility capable of producing enough electricity for over 1 million homes. The plant site is punctuated by two natural-draft hyperbolic cooling towers, each 507 feet tall, which help cool the plant. Limerick's two boiling water reactors, designed by General Electric, are each capable of producing 1,143 net megawatts. Unit 1 began commercial operation in February 1986, with Unit 2 going on-line in January 1990.

x A(x) x

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

y Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. x So what is the volume of the nose cone?

r = the y value of the function 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. x 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

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: Area of Face Thickness . Volume of the Disk A shape rotated about the y-axis would be:

y-axis is revolved about the y-axis. Find the volume. 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.

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

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

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)