Surfaces of revolution Suppose y = f ( x ) is a continuous and positive function on [ a, b ] . Let S be the surface generated when the graph of f on [ a, b ] is revolved about the x -axis. a. Show that S is described parametrically by r ( u , v ) = 〈 u, f ( u ) cos v, f ( u ) sin v 〉, for a ≤ u ≤ b , 0 ≤ v ≤ 2 π. b. Find an integral that gives the surface area of S . c. Apply the result of part (b) to find the area of the surface generated with f ( x ) = x 3 , for 1 ≤ x ≤ 2 . d. Apply the result of part (b) to find the area of the surface generated with f ( x ) = (25 – x 2 ) 1/2 , for 3 ≤ x ≤ 4.
Surfaces of revolution Suppose y = f ( x ) is a continuous and positive function on [ a, b ] . Let S be the surface generated when the graph of f on [ a, b ] is revolved about the x -axis. a. Show that S is described parametrically by r ( u , v ) = 〈 u, f ( u ) cos v, f ( u ) sin v 〉, for a ≤ u ≤ b , 0 ≤ v ≤ 2 π. b. Find an integral that gives the surface area of S . c. Apply the result of part (b) to find the area of the surface generated with f ( x ) = x 3 , for 1 ≤ x ≤ 2 . d. Apply the result of part (b) to find the area of the surface generated with f ( x ) = (25 – x 2 ) 1/2 , for 3 ≤ x ≤ 4.
Solution Summary: The author explains that the surface S is generated when the graph of f on left[a,bright] is revolved about the x -axis, the center of the circle will be on
Surfaces of revolution Suppose y = f(x) is a continuous and positive function on [a, b]. Let S be the surface generated when the graph of f on [a, b] is revolved about the x-axis.
a. Show that S is described parametrically by r(u, v) = 〈u, f(u) cos v, f(u) sin v〉, for a ≤ u ≤ b, 0 ≤ v ≤ 2 π.
b. Find an integral that gives the surface area of S.
c. Apply the result of part (b) to find the area of the surface generated with f(x) = x3, for 1 ≤ x ≤ 2.
d. Apply the result of part (b) to find the area of the surface generated with f(x) = (25 – x2)1/2, for 3 ≤ x ≤ 4.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
2.Proof that any tangent plane for the surface F(
F)
point
=
0 passses through a fixed
Sketch the surface of f(x,y)
Calculus
Assume that the level surface equation x3+y3+z3+6xyz = 1 defines z implicitly as a function of x and y. Find zx(0, −1) and zy(0, −1). Use that information to find the equation of the plane tangent to the given level surface at the point corresponding to x = 0 and y = −1−1.
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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