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6.2 - Volumes Roshan
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What is Volume? What do we mean by the volume of a solid? How do we know that the volume of a sphere of radius r is 4πr 3 /3 ? How can we give a precise definition of volume? Roshan
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Familiar Cylinders Circular cylinder –Base is a circle of radius r, and so has area πr 2 –Thus V = πr 2 h Rectangular box –Base is a rectangle of area lw –Thus V = lwh The base of a cylinder need not be circular! Roshan
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What About Other Solids? What if a solid S is not a cylinder? We can use our knowledge of cylinders: –Cut S into thin parallel slices –Treat each piece as though it were a cylinder, and add the volumes of the pieces –The thinner we make the slices, the closer we will be to the actual volume of S This leads to the method of finding volumes by cross-sections Roshan
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The Method of Cross-Sections Intersect S with a plane P x perpendicular to the x-axis Call the cross-sectional area A(x) A(x) will vary as x increases from a to b Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Roshan
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Cross-Sections The i th slab is roughly a cylinder w/ volume –Here is the base of the “cylinder” andis its height Adding the volumes of the individual slabs gives an approximation to the volume of S: Roshan
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Now we use more and more slabs –This corresponds to letting n ∞ The larger the value of n, the closer each slab becomes to an actual cylinder –In other words, our approximation becomes better and better as n ∞ So we define the volume of S by Precise Definition of Volume Roshan
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Proving the Volume of a Cone Roshan
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Example 1 Show the area of a sphere is 4πr 3 /3 Roshan
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A Special Case A solid of revolution is formed by rotating a plane region about an axis Roshan
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About Solids of Revolution The slabs are always in one of two shapes: –Either the shape of a disk… …as was the case in the sphere problem above –or the shape of a washer A washer is the region between two concentric circles Roshan
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Example 2 Find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 1. Roshan
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Find the volume of the solid obtained by rotating the region bounded by y = x 3, y = 8, and x = 0 about the y-axis NOTE: slices are perpendicular to the y-axis, rather than the x-axis Example 3 Roshan
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Example 4 Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = x and y = x 2 Roshan
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Example 5 Find the volume of the solid obtained by rotating the same region as in the preceding example BUT about the line y = 2 instead of the x-axis Roshan
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Example 6 Find the volume of the solid obtained by rotating the region in example 4 about the line x = -1 Roshan
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The formula can be applied to any solid for which the cross-sectional area A(x) can be found This includes solids of revolution, as shown above but includes many other solids as well A Bigger Picture Roshan
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Example 7a Roshan
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Example 7b Roshan Let’s take our same region but this time… The region R is the base of a solid and cross sections perpendicular to the x-axis are –a) semi-circles –b) rectangles with height = 5
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A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid. Example 8 Roshan
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Sometimes the disk/washer method is difficult for a solid of revolution An alternative is to divide the solid into concentric circular cylinders This leads to the method of cylindrical shells Method of Cylindrical Shells Roshan
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Example of Cylindrical Shells Problem: Suppose the region bounded by y = 2x 2 – x 3 and the x-axis, is rotated about the y-axis To use washers, we must find the inner and outer radii, which would require solving a cubic equation Shells Method: We can find this volume by rotating an approximating rectangle about the y-axis Note that our cross section is now parallel to the axis we are revolving about Roshan
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