Determination of Curvilinear Areas. The Parabola

Excerpts

• Determination of Curvilinear Areas. The Parabola

  Hence the curvilinear area $MPP'M'$ is the limit towards which we continually approach, but which we never reach, by dividing $MM'$ into a greater and greater number of equal parts, and adding the parallelograms $Mr$, $mr'$, etc., so obtained.

• Determination of Curvilinear Areas. The Parabola

  These are the altitudes of a set of parallelograms, the base of each of which is $dx$; hence the sum of their area is

• Determination of Curvilinear Areas. The Parabola

  If we take the function $cx^{n}$, $c$ being independent of $x$, and substitute $x + h$ for $x$, we have for the development $cx^{n} + cnx^{n-1}\, h + \etc$.

• Determination of Curvilinear Areas. The Parabola

  Here $y = p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, and we must find the integral of $p^{\efrac{1}{2}} x^{\efrac{1}{2}}\, dx$, or the function whose differential coefficient is $p^{\efrac{1}{2}} x^{\efrac{1}{2}}$, $p^{\efrac{1}{2}}$ being a constant.

• Determination of Curvilinear Areas. The Parabola

  Hence, $y$ being the ordinate, the area included between the axis of $x$, any two values of $y$, and the portion of the curve they cut off, is $\int y\, dx$, beginning at the one ordinate and ending at the other.

• Determination of Curvilinear Areas. The Parabola

  at a time when such a step was one of no small magnitude.

Equations

Problems

No exercises in this chapter.