CentroidIn Exercises 47-52, find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple
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Chapter 14 Solutions
Calculus: Early Transcendental Functions
- Finding the Volume of a Solid In Exercises 17-20, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4.y =1/2x3, y = 4, x = 0arrow_forwardA). Use Pappus's theorem for surface area and the fact that the surface area of a sphere of radius d is 4pid^2 to find the centroid of the semicircle x=(d^2-y^2)^0.5arrow_forwardusing 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_forward
- A lamina occupies the part of the disk x2+y2≤a2x2+y2≤a2 that lies in the first quadrant. Find the center of mass of the lamina if the density function is p(x,y)=xy2arrow_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_forwardVariable-density solids Find the coordinates of the center of mass of the following solid with variable density. The region bounded by the paraboloid z = 4 - x2 - y2 andz = 0 with ρ(x, y, z) = 5 - zarrow_forward
- Center of mass of constant-density solids Find the center of mass of the following solid, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The tetrahedron in the first octant bounded by z = 1 - x - y andthe coordinate planesarrow_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_forwardConsider the solid bounded by z = 7 − x, x = 6 − 4y, x = 4y, x = 0, and z = 0 with density function δ(x,y,z) = 17xy kg/m3 and length measured in meters. a) Draw a graph of the given region and include it with your written work. b) Set up and evaluate a triple integral to determine the mass of the solid. c) Determine the center of mass of the solid in rectangular coordinates.arrow_forward
- IntegrationDetermine the volume of the solid below the paraboloid z=x²+3y² and above the region bounded by the planes x=0 ,y=1,y=x and z=0arrow_forwardUse a triple integral to find the volume of the solid; the solid lies in the first octant bounded by the coordinate planes and the plane 3x+6y+4z=12.arrow_forwardSet up but DO NOT EVALUATE a triple integral to find the volume of the tetrahedron with vertices (0,0,0), ( 6, 0, 0), (0, 3, 0) and (0, 0, 4). one question two partsarrow_forward
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