Concept explainers
Moments of Inertia In Exercises 57 and 58, verify the moments of inertia for the solid of uniform density. Use a computer algebra system to evaluate the triple
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Multivariable Calculus
- 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_forwardUsing 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_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
- Variable-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_forwardTrue or False Plus A. In evaluating the moment of a planar lamina, a horizontal strip cannot be used as a representative area. B. The moment of any planar lamina is the product of the mass of the region and its centroid. Choices A. Both A and B are true B. Both A and B are false C. A is true, B is false D. A is false, B is truearrow_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
- find a. the mass of the solid. b. the center of mass. A solid region in the first octant is bounded by the coordinate planes and the plane x + y + z = 2. The density of the solid is d(x, y, z) = 2x gm/cm3arrow_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_forwardUsing Stokes’ Theorem to evaluate a surface integral Evaluate∫∫S (∇ x F) # n dS, where F = -y i + x j + z k, in the following cases.a. S is the part of the paraboloid z = 4 - x2 - 3y2 that lies within the paraboloid z = 3x2 + y2 (the blue surface as shown). Assume n pointsin the upward direction on S.b. S is the part of the paraboloid z = 3x2 + y2 that lies within the paraboloidz = 4 - x2 - 3y2, with n pointing in the upward direction on S.c. S is the surface in part (b), but n pointing in the downward direction on S.arrow_forward
- A lamina occupies the part of the disk x2 + y2 ≤ 16 in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.arrow_forwardUsing geometry, calculate the volume of the solid under z=√49−x2 −y2 and over the circular disk x2 +y2 ≤49.arrow_forwardFlux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning