13 years ago
Anonymous

Find the volume of the described volume S?

The base of S is a circular disk with the radius r. Parallel cross sections perpendicular to the base are squares.
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13 years ago
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You must be in calculus. First you need to visualize what your object looks like. The bottom is a circle, and at the center, it is as high as the diameter, so if you cut it in two perpendicular to the base you have a square cross section. In fact anywhere you cut it you have a square. This is key. If you can figure out the height curve (which is equal to the chord of circle parallel to the diameter at any given point, then you can integrate to find the area under the curve. So here goes. Draw a circle, and draw the diameter vertically through it. Then draw another horizontally through it. Treat the horizontal line as the x-axis. Pick a point on the circle's edge and draw the radius to it. Also, drop a line down to the x-axis. Now you have a right triangle and the end of the hypotenuse is your point. The length of the line from the point to the x-axis is given by (r)(sin α) where α = the angle between the radius and the x-axis. So, the length of the chord is 2r sin α. We know that the height of the object is this same length since the cross section is square, so the area of that cross section is (2r sin α)². If you integrate (2r sin α)² from α = 0 to 90, then you will have the volume of half teh object. Multiply it by two for your answer. I hope that helps, and I hope it is correct.... it has been a while. Either way it should get you moving on the right track. Good luck.
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