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Math 1B § 6.2 Volumes Overview: The volume of a solid can be computed by “integrating its cross sectional area”. Think of slicing the solid with a knife through x and computing the area of this slice. b Volume = ∫ A( x )dx a Here’s the reason: Approximate the volume by slicing it into thin cylindrical “slabs”. € Adding the volumes of these slabs, we get an approximation for the total volume. n Volume ≈ ∑ A( x *i )Δx i=1 The approximation gets better as the slabs get thinner (as n → ∞ ), so € n Volume = lim ∑ A( x *i )Δx n →∞ i=1 But we recognize the right side as a definite integral. Therefore, € b Volume = ∫ A( x )dx a Stewart – 7e € 1 Solids of Revolution Let f be nonnegative and continuous on [ a,b] and let R be the region bounded above by the graph of f , below by the x-axis, and on the sides by x = a and x = b . € y = f(x) R a b When this region is revolved about the x-axis, it generates a solid having circular cross sections. Since the cross section at x has radius f ( x ) , its area is A ( x ) = Volume = Since the cross sections are disk shaped, this application is called the method of disks. Example: Find the volume of the solid obtained when the region under the curve y = x 2 from 0 to 1 is revolved about the x-axis. Stewart – 7e 2 Example: Find the volume of the solid obtained when the region enclosed by y = 1 , y = 1 , y = 2 , and x the y-axis is revolved about the y-axis. Stewart – 7e 3 Suppose f and g are nonnegative continuous functions on [ a,b] such that g( x ) ≤ f ( x ) . Let R be the region bounded by the graphs of f and g, x = a , and x = b . € R € When this region is revolved about the xaxis, it generates a solid having washer shaped cross sections. Since the cross section at x has inner radius g ( x ) and outer radius f ( x ) , its area is A ( x) = Volume = This application is called the method of washers. Stewart – 7e 4 Example: Revolve the region between the graphs of y = x 2 and y = x3 about the x-axis. Find the volume of the resulting solid. Stewart – 7e 5 Example: Find the volume of the solid obtained by rotating the region in the previous example about the y-axis. Stewart – 7e 6 Example: Now rotate the same region about the line y = 1 . Stewart – 7e 7 Example: Now rotate the same region about the line x = 2 . Stewart – 7e 8