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.
Let f be a function whose graph consists of 5 line segments and a semicircle as shown in the figure below.
Let g(x) = √ƒƒ(t) dt .
0
3
2
-2
2
4
5
6
7
8
9
10
11
12
13
14
15
1. g(0) =
2. g(2) =
3. g(4) =
4. g(6) =
5. g'(3) =
6. g'(13)=
The expression 3 | (3+1/+1)
of the following integrals?
A
Ов
E
+
+
+ +
18
3+1+1
3++1
3++1
(A) √2×14 dx
x+1
(C) 1½-½√ √ ² ( 14 ) d x
(B) √31dx
(D) So 3+x
-dx
is a Riemann sum approximation of which
5
(E) 1½√√3dx
2x+1
2. Suppose the population of Wakanda t years after 2000 is given by the equation
f(t) = 45000(1.006). If this trend continues, in what year will the population reach 50,000
people? Show all your work, round your answer to two decimal places, and include units. (4
points)
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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