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Unformatted text preview: Week 6 Outline Compute the area of a surface of revolution –Surface Area for Revolution about the xaxis –Surface Area for Revolution about the yaxis Center of Mass – Masses Along a LineOne Dimensional Discrete Case – Wires and Thin RodsOne Dimensional Continuous Case – Masses Distributed over a Plane RegionTwo Dimensional Discrete Case – Thin, Flat PlatesTwo Dimensional Continuous case Parametric curves Plane curves and parametrizations A surface of revolution is a threedimensional surface with circular cross sections, like a vase. To find the area of a surface of revolution, we can divide the surface into narrow circular bands, figure the surface area of a representative band, and then just add up the areas of all the bands to get the total surface area. Surface Area for Revolution about the xaxis If the function f ( x ) ≥ 0 is continuously differentiable on [ a,b ], the area of the surface generated by revolving the curve y = f ( x ) about the xaxis is S = Z b a 2 πf ( x ) p 1 + ( f ( x )) 2 d x. Surface Area for Revolution about the yaxis If x = g ( y ) ≥ 0 is continuously differentiable on [ c,d ], the area of the surface generated by revolving the curve x = g ( y ) about the yaxis is S = Z d c 2 πg ( y ) p 1 + ( g ( y )) 2 d y. Examples 1. Find the area of the curved surface of a rightcircular cone of base radius r and height h by rotating the straight line segment from (0 , 0) to ( r,h ) about the yaxis. 1 2. A band is formed by being cut from a sphere of radius R by parallel planes h units apart. Compute the surface area of the band. 3. A disk of radius a has center at the point ( b, 0), where b > a > 0. The disk is rotated about the y axis to generate a torus. Find its surface area. 4. Find the surface area of the surface obtained by rotating the ellipse x 2 + 4 y 2 = 4 about the xaxis. 5. Find the surface area of the surface obtained by rotating the ellipse x 2 + 4 y 2 = 4 about the yaxis. 6. For what values of k does the surface generated by rotating the curve y = x k , < x < 1, about the yaxis have a finite surface area? 7. A hollow container in the shape of an infinitely long horn is generated by rotating the curve y = 1 x , 1 ≤ x < ∞ , about the xaxis. (a) Find the volume of the container. (b) Show that the container has infinite surface area. (c) How do you explain the paradox that the container can be filled with a finite volume of paint but requires infinitely much paint to cover its surface? Masses Along a LineOne Dimensional Discrete Case Suppose that masses m 1 , m 2 ,..., m k are on a rigid xaxis supported by a fulcrum at the origin. Let Torque of m i = m i gx i , i = 1 , 2 ,...,k, where x i is the signed distance from the point of application to the origin. This measures the turning effect by mass x i . Masses to the left of the origin exert negative (counter....
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 Fall '06
 GROSS
 Calculus, Parametric surface, y axis Center of Mass

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