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n X f xk 2 yk 2 g1=2 k 1
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or equivalently lim ( n X
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where xk xk xk 1 and yk yk yk 1 . Thus, the length of the arc of a curve in rectangular Cartesian coordinates is b ( 2  2 )1=2 dx dy 0 2 0 2 1=2 L f f t g t g dt dt dt dt a (This form may be generalized to any number of dimensions.) Upon changing the variable of integration from t to x we obtain the planar form ! )1=2 f b ( dy 2 1 L dx f a (This form is only appropriate in the plane.) The generic di erential formula ds2 dx2 dy2 is useful, in that various representations algebraically arise from it. For example, ds dt expresses instantaneous speed.
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AREA Area was a motivating concept in introducing the integral. Since many applications of the integral are geometrically interpretable in the context of area, an extended formula is listed and illustrated below. Let f and g be continuous functions whose graphs intersect at the graphical points corresponding to x a and x b, a < b. If g x A f x on a; b , then the area bounded by f x and g x is b A fg x f x g dx
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If the functions intersect in a; b , then the integral yields an algebraic sum. g x sin x and f x 0 then: 2 2  sin x dx cos x 0 
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VOLUMES OF REVOLUTION Disk Method Assume that f is continuous on a closed interval a @ x @ b and that f x A 0. Then the solid realized through the revolution of a plane region R (bound by f x , the x-axis, and x a and x b) about the x-axis has the volume b V  f x 2 dx
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INTEGRALS
This method of generating a volume is called the disk method because the cross sections of revolution are circular disks. (See Fig. 5-5(a).)
Fig. 5-5 EXAMPLE. A solid cone is generated by revolving the graph of y kx, k > 0 and 0 @ x @ b, about the x-axis. Its volume is  b 3 3 k3 x3 b  k b V  k2 x2 dx   3 0 3 0
Shell Method Suppose f is a continuous function on a; b , a A 0, satisfying the condition f x A 0. Let R be a plane region bound by f x , x a, x b, and the x-axis. The volume obtained by orbiting R about the y-axis is b V 2x f x dx
This method of generating a volume is called the shell method because of the cylindrical nature of the vertical lines of revolution. (See Fig. 5-5(b).)
EXAMPLE. If the region bounded by y kx, 0 @ x @ b and x b (with the same conditions as in the previous example) is orbited about the y-axis the volume obtained is  b x 3 b b3 V 2 x kx dx 2k  2k  3 0 3 0
By comparing this example with that in the section on the disk method, it is clear that for the same plane region the disk method and the shell method produce di erent solids and hence di erent volumes. Moment of Inertia Moment of inertia is an important physical concept that can be studied through its idealized geometric form. This form is abstracted in the following way from the physical notions of kinetic energy, K 1 mv2 , and angular velocity, v !r. (m represents mass and v signi es linear velocity). Upon 2 substituting for v K 1 m!2 r2 1 mr2 !2 2 2 When this form is compared to the original representation of kinetic energy, it is reasonable to identify mr2 as rotational mass. It is this quantity, l mr2 that we call the moment of inertia. Then in a purely geometric sense, we denote a plane region R described through continuous functions f and g on a; b , where a > 0 and f x and g x intersect at a and b only. For simplicity, assume g x A f x > 0. Then
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