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- Find the area of the surface. The part of the cylinder y2 + z2 = 9 that lies above the rectangle with vertices (0, 0), (7, 0), (0, 2), and (7, 2).Pigeonhole Principle: consider the 3D grid with integer coordinates (That is, a point (x, y, z) lies on the grid if and only if x, y and z are all integral). Prove that if we take nine points on the grid then there exist two of the points whoseaverage is also a point on the grid.Point W is the centroid of triangle TRV. VX = 204 and RW = 104. The length of RY = ____.
- Prove that the centroid of any triangle is located at the pointof intersection of the medians. [Hints: Place the axes so thatthe vertices are (a,0) ,(0 ,b ) and (c,0). Recall that a medianis a line segment from a vertex to the midpoint of the oppositeside. Recall also that the medians intersect at a pointtwo-thirds of the way from each vertex (along the median)to the opposite side.]Find and/or verify the centroid of the common region used in engineering. Show that the centroid of the parallelogram with vertices (0, 0), (a, 0), (b, c), and (a + b, c) is the point of intersection of the diagonalsFind the area of the surface. The part of the surface z = 8 + 5x + 3y2 that lies above the triangle with vertices (0, 0), (0, 1), (2, 1).
- Find the centroid of the trapezoid with vertices (0, 0), (0, a), (c, b), and (c, 0). Show that it is the intersection of the line connecting the midpoints of the parallel sides and the line connecting the extended parallel sides, as shown in the figure.Find the centroid (¯x,¯y) of the triangle with vertices at (0,0)(1,0) and (0,5) x¯= y¯=The area of the region is bounded by the x-axis, the curve y = 6x – x2, the line x – 1 = 0 and the line x = 4. 3. Find the ordinate of the vertex of the curve y = 6x – x2. Find the area of the region. Find the distance of the centroid of the area from the y-axis.
- Find the points on the cone z2 = x2 + y2 that are closest to the point (2, 2, 0). (x, y, z) = (smaller z-value) (x, y, z) = (larger z-value)Find the volumes of the regions. The region cut from the cylinder x2 + y2 = 4 by the plane z = 0 and the plane x + z = 3Find the volumes of the regions. The region in the first octant bounded by the coordinate planes and the surface z = 4 - x2 - y