Determine whether the following sequence converges or diverges. an = {2arctan(n) n + 7} If the sequence is convergent, state the limit of the sequence. If the sequence is convergent, state the limit of the sequence. In the case of a convergent sequence, write your answer as an integer or a reduced fraction. If the solution involves a square root of a number, type sqrt in front of the number. For example, if the solution involves 2 , then type sqrt(2) for the corresponding part of the solution. If the sequence is divergent, type DIV.
(5 points) Consider the (unbounded) region R in the first quadrant consisting of the area under the curve y = 1 and to the left of the curve x = ln(e/y) (see diagram below). Let V be the volume of the solid generated by revolving R about the y-axis. Set up two expressions for V , one that uses the slicing/washer method and one that uses the cylindrical shells method. Choose one of these methods to determine the exact value of V . Show all your work (note that you need to compute an improper integral).
(1 point) In the graph below, the function f(x) is graphed with a bold, blue curve, and the function g(x) with a light, red curve. Assume that the behavior of both functions as x → ∞ is accurately suggested by the domain on which they are graphed in the figure. Suppose ∫ a ∞ f(x)dx converges. What does this graph suggest about the convergence of ∫ a ∞ g(x)dx? A. ∫ a ∞ g(x)dx diverges B. the graph does not provide enough information to suggest with any certainty whether ∫ a ∞ g(x)dx converges or diverges C. ∫ a ∞ g(x)dx neither converges nor diverges D. ∫ a ∞ g(x)dx converges
Using the disk or washer method as appropriate (Not the shell method). Find an expression for the volume of the solid that results by rotating the shaded region a) About the x-axis b) About the y-axis c) About the line y = −1 d) About the line x = a
In the figure below, the blue curve is the graph of a function h . The red curve is the graph of g(x) = (1+x) −2. Determine if the improper integral converges or diverges or if it can not be determined. Make sure to justify your answer. ∫ 5 ∞ h(x)dx (Write CONVERGES if it converges, DIVERGES if it diverges, NONE if it can not be determined)
Use the disk method to find the volume of the solid generated by revolving about the y -axis the region (shown below) enclosed by the curve x = e −y, the x -axis and the vertical line x = ln2.
Use the Shell Method or Disk Method to find the volume V of rotation generated by the region in the figure rotated about y = − 6. (Express numbers in exact form. Use symbolic notation and fractions where needed.)
Using the disk method, determine the volume of a solid formed by revolving the region bounded above by the curve y = e x + 1, on the left by the line x = 0, on the right by the line x = 1, and below by the line the y = 0 about the x -axis. The 2 d picture below may help in determining the radius of the disk used in setting up the integral for the volume. For a dynamic 3d look at the solid, click here (This will open a new window.) Part 1. Setup the integral that represents the volume of the solid of revolution described above.. = ∫
(1 point) Finding the volume of a solid of revolution (disk method) Using the disk method, determine the volume of a solid formed by revolving the region bounded above by the line y = 16, on the left by the line x = 0, and below by the curve the x = y 3/2 about the line y = 16. The 2d picture below may help in determining the radius of the disk used in setting up the integral for the volume. For a dynamic 3d look at the solid, click here (This will open a new window.) Part 1. Setup the integral that represents the volume of the solid of revolution described above.. = ∫ Part 2. The volume of the solid is units cubed. NOTE: Type an exact value without using decimals. Note: You can earn partial credit on this problem.
Used the disk method to find the volume of the cone generated by revolving about the x -axis the region (shown below) enclosed by the curve y = 2x, the x -axis and the vertical line x = 1. Confirm your result using the formula for the volume of a cone ( V = 1/3πr2h).
In the graph below, the function f(x) is graphed with a bold, blue curve, and the function g(x) with a light, red curve. Assume that the behavior of both functions as x → ∞ is accurately suggested by the domain on which they are graphed in the figure. Suppose ∫ a ∞ f(x)dx converges. What does this graph suggest about the convergence of ∫ a ∞ g(x)dx? A. ∫ a ∞ g(x)dx converges B. the graph does not provide enough information to suggest with any certainty whether ∫ a ∞ g(x)dx converges or diverges C. ∫ a ∞ g(x)dx neither converges nor diverges D. ∫ a ∞ g(x)dx diverges
Use either the shell method or the disk/washer method to find the volume of the solid of revolution generated by revolving the shaded region bounded by the graphs of f(x) = −x2 + 21 and g ( x ) = 8x + 1 and the y -axis about the x -axis. The graph is not drawn to scale. The graphs f and g intersect at (2,17). (Express numbers in exact form. Use symbolic notation and fractions where needed.) volume:
In the graph below, the function f(x) is graphed with a bold, blue curve, and the function g(x) with a light, red curve. Assume that the behavior of both functions as x → ∞ is accurately suggested by the domain on which they are graphed in the figure. Suppose ∫ a ∞ f(x)dx converges. What does this graph suggest about the convergence of ∫ a ∞ g(x)dx? A. ∫ a ∞ ∞ g(x)dx converges B. ∫ a ∞ g(x)dx diverges C. the graph does not provide enough information to suggest with any certainty whether ∫ a ∞ g(x)dx converges or diverges D. ∫ a ∞ g(x)dx neither converges nor diverges
Consider the series ∑ n = 1 ∞ an = ∑ n = 1 ∞ 3 qn (qn) 3 n8 n + 2. To determine whether this series converges, it is convenient to use the ratio test. Using this test, we can express the quotient |a n + 1/an| of any two subsequent terms in the form a q⋅(b + c n) 3⋅n8 n + 2 (n + 1)8 n + 1 + 2, where a = b = and c = The series will converge for q < Finally, we want to check whether the series converges with q at the upper bound of this range (since the test we just applied is inconclusive there). We compare with the p -series ∑ n = 1 ∞ bn, where b n = K 1 n p , with K is a constant and p = , and so when q is equal to the upper bound previously identified the series is (type "c" or "d" for convergent or divergent, respectively).
In the graph below, the function f(x) is graphed with a bold, blue curve, and the function g(x) with a light, red curve. Assume that the behavior of both functions as x → ∞ is accurately suggested by the domain on which they are graphed in the figure. Suppose ∫ a ∞ f(x)dx converges. What does this graph suggest about the convergence of ∫ a ∞ g(x)dx? A. the graph does not provide enough information to suggest with any certainty whether ∫ a ∞ g(x)dx converges or diverges B. ∫ a ∞ g(x)dx diverges C. ∫ a ∞ g(x)dx neither converges nor diverges D. ∫ a ∞ g(x)dx converges
Find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. ∥u∥ = 5, θu = 0 ∘ ∥v∥ = 2, θ v = 60 ∘ u + v =
Three forces with magnitudes of 60 pounds, 99 pounds, and 143 pounds act on an object at angles of 30∘, 45∘, and 120∘ respectively, with the positive x -axis. Find the direction and magnitude of the resultant force. (Round your answers to one decimal place.) direction magnitude Ib
A gun with a muzzle velocity of 1000 feet per second is fired at an angle of 13∘ above the horizontal. Find the vertical and horizontal components of the velocity. (Round your answers to two decimal places.) horizontal component ft/sec vertical component ft/sec
Find the lengths of the sides of the triangle ABC with the indicated vertices. (Round your answer to three decimal places.) A(5, 0, 0) B(0, 7, 0) C(0, 0, − 12) |AC| = |BC| = Determine whether the triangle ABC is a right triangle, an isosceles triangle, or neither. right triangle isosceles triangle neither
The guy wire supporting a 100-foot tower has a tension of 600 pounds. Using the distances shown in the figure, write the component form of the vector F representing the tension in the wire. (Round your answers to three decimal places.) F =
Find the tension in each of the supporting cables in the figure if the weight of the crate is 380 newtons. (Round your answers to three decimal places.) tension in AB → N tension in AC → N tension in AD → N
Determine whether u and v are orthogonal, parallel, or neither. u = ⟨cos(θ), sin(θ), 6⟩ v = ⟨sin(θ), −cos(θ), 0⟩ parallel orthogonal neither
Find the direction cosines and angles of u and show that cos2α + cos2β + cos2γ = 1. (Round your answers for the angles to four decimal places.) u = −4i + 3j + 9k cosα = α ≈ rad cosβ = β ≈ rad cosγ = γ ≈ rad