Areas bounded by curves. Work diagrams

—The area enclosed between two different curves. Let PABQ and PA B Q (Fig. 103) be two curves, it is required to find the area PABQB A1. Let yl = fx(x) be the equation of one curve, y2 = f2(x), the equation of the other. First find the abscissae of the points of intersection of the two curves. Find separately the 

To find the area of the portion ABB Alet xl be the abscissa of AB and x2 the abscissa of BS, then, 

In illustration let us consider the area included between the two parabolas whose equations are y2 = Ax and x2 = 4y. The curves obviously meet at the origin, and at the point x = 4, cm., say, y = 4 cm. (16), page 95. Consequently, 

Let a given volume, x, of a gas be contained in a cylindrical vessel in which a tightly fitting piston can be made to slide (Fig. 

Let the sectional area of X the piston be unity. Now let the volume of. gas change dx units when a slight pressure X is applied to the free end of the piston. Then, by definition of work, W, 

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