(b) Find a parameterization, in spherical coordiantes, for the ellipsoid 4x2 + 9y2 + 2²/4 = 1 that lies above the xY-plane.
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- Suppose that a cylindrical container of radius r and height L is filled with a liquid with volume V , and rotated along the y-axis with constant angular speed ω. This makes the liquid rotate, and eventually at the same angular speed as the container. The surface of the liquid becomes convex as the centrifugal force on the liquid increases with the distance from the axis of the container. The surface of the liquid is a paraboloid of revolution generated by rotating the parabola y = h + ω2x2/2g around the y-axis, where g is gravitational acceleration and h is shown below. (You can take g=32ft/s2 or 9.8m/s2). Express h as a function of ω. (2) At what angular speed ω will the surface of the liquid touch the bottom? At what speed will it spill over the top? (3) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed ω. The surface of the liquid is 5 ft below the top of the tank at the central…Consider the ellipse E in the xy-plane defined by the equation ax2 + y2 = 1 where a is positive. (1) Find a parametrization r(t) of E (2) Find all the points where r(t) is orthogonal to r'(t).find a parametrization for the curve. the lower half of the parabola x - 1 = y2
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- Graph the curve 8y2 = x2 (1 − x2 ). Use a computer algebra system to find the surface area of the solid of revolution obtained by revolving the curve about the y-axis.Find the point on the ellipsoid x2 + y2/4 + z2/9 = 1 for which x + y + z is largest.Find a generating curve and the axis of revolution for the surface x2 + 3y2 + z2 = 9.