An Elementary Treatise On Variable Quantities, in Two Parts: the Direct And Inverse
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70. There can be only one rate of a given function, but there may be more than one function answering to a given rate. This is obvious, since x'^ and x'^ -\- a have the same rate, viz., 2xdx. Therefore, in integrating, a constant term must be added to the integral. This term is usually represented by C ; thus the integral of du = 2axdx is u^^ax -\- C.
C is called an arbitrary constant, and the integral before the value of C is known is called an incomplete integral. In the solution of a real pr...oblem, however, the value of C may be determined from the known conditions of the problem and consequently a complete integral obtained.
For illustration take the ratal equation of the straight line, dy = adx, whence y=^ax-\-C.
If the straight line passes through the origin of the coordi- nates, then y = when x = 0; hence C ^ 0, and the complete or true integral is y = ax.
102 AN ELEMENTARY TREATISE But if the straight line cuts the axis of ordinates at a dis- tance from the origin equal to b, then for x = 0, y = b, conse- quently C = b, and the true integral is y^ ax -{- b.
What to read after An Elementary Treatise On Variable Quantities, in Two Parts: the Direct And Inverse?
You can find similar books in the "Read Also" column, or choose other free books by Cook, Hiram, 1827-1917 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
70. There can be only one rate of a given function, but there may be more than one function answering to a given rate. This is obvious, since x'^ and x'^ -\- a have the same rate, viz., 2xdx. Therefore, in integrating, a constant term must be added to the integral. This term is usually represented by C ; thus the integral of du = 2axdx is u^^ax -\- C.
C is called an arbitrary constant, and the integral before the value of C is known is called an incomplete integral. In the solution of a real pr...oblem, however, the value of C may be determined from the known conditions of the problem and consequently a complete integral obtained.
For illustration take the ratal equation of the straight line, dy = adx, whence y=^ax-\-C.
If the straight line passes through the origin of the coordi- nates, then y = when x = 0; hence C ^ 0, and the complete or true integral is y = ax.
102 AN ELEMENTARY TREATISE But if the straight line cuts the axis of ordinates at a dis- tance from the origin equal to b, then for x = 0, y = b, conse- quently C = b, and the true integral is y^ ax -{- b.
What to read after An Elementary Treatise On Variable Quantities, in Two Parts: the Direct And Inverse?
You can find similar books in the "Read Also" column, or choose other free books by Cook, Hiram, 1827-1917 to read onlineMoreLess
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