Concluding Remarks on the Study of the Calculus

Excerpts

• Concluding Remarks on the Study of the Calculus

  Thus, if he has the area of a curve to find, instead of merely saying that $y$, the ordinate, being a certain function of the abscissa $x$, $\int y\, dx$ within the given limits would be the area required;

• Concluding Remarks on the Study of the Calculus

  let him remark that if an approximate solution only were required, it might be obtained by dividing the curvilinear area into a number of four-sided figures, as in [Figure]10, one side of which only is curvilinear, and embracing so small an arc that it may, without visible error, be considered as rectilinear.

• Concluding Remarks on the Study of the Calculus

  The mathematical method begins with the same principle, investigating upon this supposition, not the sum of these rectilinear areas, but the limit towards which this sum approaches, as the subdivision is rendered more minute.

• Concluding Remarks on the Study of the Calculus

  This limit is shown to be that of which we are in search, since it is proved that the error diminishes without limit, as the subdivision is indefinitely continued.

• Concluding Remarks on the Study of the Calculus

  The method so generally followed in our elementary works, of leading the student at once into the mechanical processes of the science, postponing entirely all other considerations, is to many students a source of obscurity at least, if not an absolute impediment to their progress; since they cannot imagine what is the object of that which they are required to do.

Equations

Problems

No exercises in this chapter.