121616949-math.202

121616949-math.202 - 188 Chapter 9 Applications of...

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188 Chapter 9 Applications of Integration Of course a real “slice” of this figure will not have straight sides, but we can approxi- mate the volume of the slice by a cylinder or disk with circular top and bottom and straight sides; the volume of this disk will have the form πr 2 Δ x . As long as we can write r in terms of x we can compute the volume by an integral. EXAMPLE 9.9 Find the volume of a right circular cone with base radius 10 and height 20. (A right circular cone is one with a cicular base and with the tip of the cone directly over the center of the base.) We can view this cone as produced by the rotation of the line
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Unformatted text preview: 9.9 . 20 ............................................................................................................... Figure 9.9 A region that generates a cone; approximating the volume by circular disks. ( JA ) At a particular point on the x-axis, say x i , the radius of the resulting cone is the y-coordinate of the corresponding point on the line, namely y i = x i / 2. Thus the total volume is approximately n-1 X i =0 π ( x i / 2) 2 dx and the exact volume is Z 20 π x 2 4 dx = π 4 20 3 3 = 2000 π 3 . Note that we can instead do the calculation with a generic height and radius: Z h π r 2 h 2 x 2 dx = πr 2 h 2 h 3 3 = πr 2 h 3 ,...
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