Moments of Inertia In Exercises 53- 56, find
(a)
(b)
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Chapter 14 Solutions
Calculus (MindTap Course List)
- using calculus Find the center of mass of the region bounded by the following functions.(a) y = 0, x = 0, y = ln x and x = e(b) y = 2√x and y = x(c) y = sin x, y = cos x, x = 0, and x = π/4.arrow_forwardElectric charge is distributed over the disk x2+y2=1 so that its charge density is σ(x,y)= 1+x2+y2 (Kl/m2). Calculate the total charge of the disk.arrow_forwardConsider the solid E that occupies the tetrahedral region formed by the coordinate planes, x = 0, y = 0 and z = 0 and the plane (x/a) + (y/b) + (z/c) = 1 for some positive constants a, b, and c. Assume the mass density is ρ(x, y, z) = 1. Find the x-coordinate, of center of mass of the solid.arrow_forward
- Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.F = (y-x) i + (z-y) j + (z-x) k ; D: the region cut from the solid cylinder x 2 + y 2 ≤ 49 by the planes z = 0 and z=2 a) 0 b) 98π c) -98π d) -98arrow_forwardVariable-density solids Find the coordinates of the center of mass of the following solid with the given density. The cube in the first octant bounded by the planes x = 2, y = 2,and z = 2, with ρ(x, y, z) = 1 + x + y + zarrow_forwardSurface integral of a vector field? Let T be the upper surface of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 4. Calculate the integral of the image below, where S is the face of T that is in the xy plane.arrow_forward
- Set up the triple integrals required to find the center of mass of the solid tetrahedron whose density is the constant k and has vertices at (0,0,0), (2,0,0), (0,1,0), and (0,0,4). Do Not evaluate the integral, only set it up.arrow_forwardSet-up the iterated double integral in rectangular coordinates equalto the volume of the solid in the first octant bounded above by the paraboloid z = 1−x2-y2, below by the plane z =3/4, and on the sides by the planes y = x and y = 0.arrow_forwardThe area bounded by y = 3, x = 2, y = -3 and x = 0 is revolved about the y-axis. a.The x-coordinate of its centroid is… b. The y-coordinate of the centroid of the solid is… c. The moment of inertia of the solid is…arrow_forward
- (a) A triangular lamina with vertices (0,0), (-4,2), (6,2) has the density function δ(x,y) =xy i) Sketch the lamina. ii) Find the mass of the lamina. (b) Find the surface area of the portion of the paraboloid z= 2-x2-y2 above the xy-planearrow_forward(a) Find the centroid of the area between the x axis and one arch of y = sin x.(b) Find the volume formed if the area in (a) is rotated about the x axis.(c) Find Ix of a mass of constant density occupying the volume in (b).arrow_forwardSet-up the double integral to find the mass of the surface S : the part of the plane z = 3 − x − 2y in the first octant, if the mass density at any point on the surface is given by δ(x, y, z) = xz with units of mass per unit area. You do not need to evaluate the double integral.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning