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Home Applications of the Integral Volumes of Solids of Revolution Problems  
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In Problems 110 the region under the given curve is rotated about (a) the xaxis, (b) the yaxis. Sketch the region and find the volumes of the two solids of revolution. In Problems 1122 the region bounded by the given curves is rotated about (a) the xaxis, (b) the yaxis. Sketch the region and find the volumes of the two solids of revolution. In Problems 2334 the region under the given curve is rotated about the xaxis. Find the volume of the solid of revolution. In Problems 3546 the region is rotated about the xaxis. Find the volume of the solid of revolution. 47 A hole of radius a is bored through the center of a sphere of radius r (a < r). Find the volume of the remaining part of the sphere. 48 A sphere of radius r is cut by a horizontal plane at a distance c above the center of the sphere. Find the volume of the part of the sphere above the plane (c < r). 49 A hole of radius a is bored along the axis of a cone of height h and base of radius r. Find the remaining volume (a < r). 50 Find the volume of the solid generated by rotating an ellipse a^{2 }x^{2} + b^{2 }y^{2} = 1 about the xaxis. Hint: The portion of the ellipse in the first quadrant will generate half the volume. 51 The sector of a circle shown in the figure is rotated about (a) the xaxis, (b) the yaxis. Find the volumes of the solids of revolution. 52 The region bounded by the curves y = x^{2}, y = x is rotated about (a) the line y = 1, (b) the line x = 2. Find the volumes of the solids of revolution. 53 Find the volume of the torus (donut) generated by rotating the circle of radius r with center at (c, 0) around the yaxis (r < c). 54 (a) Find a general formula for the volume of the solid of revolution generated by rotating the region bounded by the curves y = f(x), y = g(x), a ≤ x ≤ b, about the line y=k. (b) Do the same for a rotation about the line x = h.


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