1120_6 - Week 6 Outline Compute the area of a surface of...

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Week 6 Outline Compute the area of a surface of revolution –Surface Area for Revolution about the x -axis –Surface Area for Revolution about the y -axis Center of Mass – Masses Along a Line-One Dimensional Discrete Case – Wires and Thin Rods-One Dimensional Continuous Case – Masses Distributed over a Plane Region-Two Dimensional Discrete Case – Thin, Flat Plates-Two Dimensional Continuous case Parametric curves Plane curves and parametrizations A surface of revolution is a three-dimensional 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 x -axis 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 x -axis is S = Z b a 2 πf ( x ) p 1 + ( f 0 ( x )) 2 d x. Surface Area for Revolution about the y -axis 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 y -axis is S = Z d c 2 πg ( y ) p 1 + ( g 0 ( y )) 2 d y. Examples 1. Find the area of the curved surface of a right-circular cone of base radius r and height h by rotating the straight line segment from (0 , 0) to ( r, h ) about the y -axis. 1
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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 x -axis. 5. Find the surface area of the surface obtained by rotating the ellipse x 2 + 4 y 2 = 4 about the y -axis. 6. For what values of k does the surface generated by rotating the curve y = x k , 0 < x < 1, about the y -axis 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 x -axis. (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 Line-One Dimensional Discrete Case Suppose that masses m 1 , m 2 ,. . . , m k are on a rigid x -axis 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- clockwise) torque. Masses to the right of the origin exert positive (clockwise) torque. Now
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