Exercise 7. In the (yz) plane consider the set A of vertices (3,0), (4, 2), (5, 0) and (4, –2). The volume of the solid obtained rotating A around the z axes is A) 327 В) 16л C) 8/27 D) 247
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A: To find the volume of the given solid.
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Q: Find the volume of the solid bounded by the sphere x2+y2+z2=6 and the paraboloid x2+y2=z.
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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a orthocenter? b centroid?Does a right circular cylinder such as an aluminum can have a symmetry with respect to at least one plane? b symmetry with respect to at least one line? c symmetry with respect to a point?
- Find the coordinates of the centroid of the triangle with the given vertices. X(1, 4), Y(7, 2), Z(2, 3)Sketch the solid in the first octant that is enclosed by the planes x = 0, z = 0, x = 5, z − y = 0, and z = −2y + 6.Determine the flux of F(x, y, z) = < −x2 + x, y, 8x3 − z + 9 > across the surface with an upward orientation. Let the surface be the portion of the paraboloid z = 9 − 4x2 −4y2 on the first octant above the plane z = 1. (note: do not use gauss' theorem)
- A rectangle ℛ with sides a and b is divided into two parts ℛ1 and ℛ2 by an arc of a parabola that has its vertex at one corner of ℛ and passes through the opposite corner. Find the centroids of both ℛ1 and ℛ2.The part of the surface z =1 + 3x+ 2y 2 that lies above the triangle with vertices (0, 0), (0, 1) and (2, 1)Prove that the centroid of a parallelogram is the point of intersection of the diagonals of the parallelogram. [Hint: Choose coordinates so that the vertices of the parallelogram are located at (0, 0), (0, a), (b, c), and (b, a + c).]