Explanation Given: Rotating the circle ( x − b ) 2 + y 2 = a 2 for b > a about the y-axis Calculation: For a smooth surface defined parametrically on the region D as follows: R ( u , v ) = x ( u , v ) i + y ( u , v ) j + z ( u , v ) k The surface area S is given by: S = ∬ D ‖ R u × R v ‖ d u d v , where R u = ∂ R ∂ u = ∂ x ∂ u i + ∂ y ∂ u j + ∂ z ∂ u k R v = ∂ R ∂ v = ∂ x ∂ v i + ∂ y ∂ v j + ∂ z ∂ v k Area Theorem of Pappus: Let C be a curve of length L in the plane Then the surface obtained by rotation of C about the axis L in the plane has area 2 π L h , where h is the distance from the centroid of C to the axis of rotation. Suppose a torus is formed by the rotation of the circle ( x − b ) 2 + y 2 = a 2 for b > a about the y-axis. Refer to the figure provided in the question. The objective is to find the surface area of the torus by Pappus area theorem and also compare the result with area of the torus that is found parametrically. The area of the torus by Pappus area theorem is as follows: The length of the curve is the circumference of the provided circle. Therefore, L = 2 π a Since, the axis of rotation is y-axis, the distance, h of the centroid from the axis of rotation is #. Therefore, by Pappus theorem, surface area of the torus equals 2 π × ( 2 π a ) × b . That is, surface area of the torus is ( 4 π 2 a b ) . The area of the torus parametrically is as follows: R ( u , v ) = ( a + cos v ) ( cos u ) i + ( a + b cos v ) ( sin u ) j + ( b sin v ) k for 0 ≤ u ≤ 2 π , 0 ≤ v ≤ 2 π , 0 ≤ b ≤ a Thus, R u , R v are given as follows: R u = ( − a sin u − b ( cos v ) sin u ) i + ( a cos u + b cos v cos u ) j + ( 0 ) k R v = ( − b sin v ) ( cos u ) i + ( − b sin v ) ( sin u ) j + ( b cos v ) k Calculate R u , R v as follows: | i j k ( − a

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ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 12.6, Problem 60PS

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To find: the surface area of the torus by applying Pappus area theorem and compare the result with the area found parametrically

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Chapter 12.6, Problem 59PS

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the surface area of the torus by applying Pappus area theorem and compare the result with the area found parametrically

Solution Summary: The author explains the Pappus area theorem and compares the result with the area found parametrically.

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To find: the surface area of the torus by applying Pappus area theorem and compare the result with the area found parametrically

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