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(c) Find the surface area of this torus. Hint: You will need to write down formulas for the surface areas of two objects. Explain what these objects are. The surface area of the torus is the sum of these two (you may as well sketch them). Do not calculate them separately, instead calculate them together, use the properties of the integral and simplify. Do not skip any steps.

(c) Find the surface area of this torus. Hint: You will need to write down formulas for the surface areas of two objects. Explain what these objects are. The surface area of the torus is the sum of these two (you may as well sketch them). Do not calculate them separately, instead calculate them together, use the properties of the integral and simplify. Do not skip any steps.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I need help with question 3. D-E please if possible.

 

3. Let C be the disk with the center at (0, R) and radius r, where R>r. If we rotate C around the x-axis we obtain an object, which is called torus. 1 (c) Find the surface area of this torus. Hint: You will need to write down formulas for the surface areas of two objects. Explain what these objects are. The surface area of the torus is the sum of these two (you may as well sketch them). Do not calculate them separately, instead calculate them together, use the properties of the integral and simplify. Do not skip any steps. (d) What is the arclength of the boundary of C? What is the center of mass of C? Which distance does the center of mass travel when C makes the full circle around the r-axis? You do not need to perform any computations here, just refer to the common knowledge and symmetry principle. (e) By comparing your calculations for the surface area of the torus with the calculations in part (d) guess The second theorem of Pappus that relates the surface area of a solid of revolution, obtained by rotating some region Q around a given axis, the arclength of the boundary of Q, and the distance travelled by the centroid of Q. Present a mathematical statement of your guess in a precise and concise form.
Transcribed Image Text:3. Let C be the disk with the center at (0, R) and radius r, where R>r. If we rotate C around the x-axis we obtain an object, which is called torus. 1 (c) Find the surface area of this torus. Hint: You will need to write down formulas for the surface areas of two objects. Explain what these objects are. The surface area of the torus is the sum of these two (you may as well sketch them). Do not calculate them separately, instead calculate them together, use the properties of the integral and simplify. Do not skip any steps. (d) What is the arclength of the boundary of C? What is the center of mass of C? Which distance does the center of mass travel when C makes the full circle around the r-axis? You do not need to perform any computations here, just refer to the common knowledge and symmetry principle. (e) By comparing your calculations for the surface area of the torus with the calculations in part (d) guess The second theorem of Pappus that relates the surface area of a solid of revolution, obtained by rotating some region Q around a given axis, the arclength of the boundary of Q, and the distance travelled by the centroid of Q. Present a mathematical statement of your guess in a precise and concise form.
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