a. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about
Question:
a. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. If C has radius r > 0 and center (R, 0, 0), show that a parametrization of the torus is
where 0 ≤ u ≤ 2π and 0 ≤ y ≤ 2π are the angles in the figure.
b. Show that the surface area of the torus is A = 4π2Rr.
r(u, v) = ((R+ r cos u)cos v)i + ((R + r cos u)sin v)j + (r sin u)k,
Step by Step Answer:
a To show that the given parametrization represents the torus we need to verify that it satisfies the conditions of a torus The parametrization of the torus is given as ru v R r cos u ...View the full answer
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir
Related Video
The area of a circle is given by the formula: A = ?r^2 Where A is the area, ? is a mathematical constant (approximately 3.14), and r is the radius of the circle. The radius is the distance from the center of the circle to the edge. So for example, if the radius of a circle is 5 units, the area would be: A = ?(5^2) = ?(25) = 78.5 square units Here three friends went out to a restaurant and decided to have pizza for lunch. While they were waiting for their order, one of them stepped out to take a call but promised to return in two minutes. The waiter came and presented them with the menu, and they decided to order two 10-inch pepperoni pizzas. The waiter took their order and left. Just then, the third friend returned and asked what they had ordered. They explained that they had ordered two small pepperoni pizzas as they were only three of them and it would be enough for them. However, the third friend suggested that for the same price, they could get a large pizza that would be much bigger than the two small ones. He asked the waiter for a pen and paper and proceeded to calculate the area of the two small pizzas and compared it to the area of one large pizza. The friends were amazed at how the circle measurements worked and decided to change their order from two small pizzas to one large pizza.
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