The place of the elementary calculus in the senior high-school mathematics : and suggestions for a modern presentation of the subject . limits is t- r2h. This ex-pressed in words is the customary rule for finding the volume of a cylinder, namely, its volumeaxis is equal to the product of the area of the base by the altitude. An Application of the the volume of a cone ofradius r and height h. The generating line has a con-stant slope. It is r/h, and it inter-sects the/ (x)-axis at the origin orzero. Then in the slope form ofthe straight line, / (x) = mx -\-b,m = r /h and b = o. Hen
The place of the elementary calculus in the senior high-school mathematics : and suggestions for a modern presentation of the subject . limits is t- r2h. This ex-pressed in words is the customary rule for finding the volume of a cylinder, namely, its volumeaxis is equal to the product of the area of the base by the altitude. An Application of the the volume of a cone ofradius r and height h. The generating line has a con-stant slope. It is r/h, and it inter-sects the/ (x)-axis at the origin orzero. Then in the slope form ofthe straight line, / (x) = mx -\-b,m = r /h and b = o. Hence in rour problem, / (x) = —x. Substitute this value in the formula,h fb fhr2 2 tt [f (x)]2 dx. This gives ir I -s-xdx. The value of this integral Ja Jo ¥ is -T-r2h. In words, this is the customary rule for finding the volume oof a cone, namely, the volume of a cone is equal to the product ofthe area of the base times one-third of the altitude. Example. Suppose that we have given a semi-circle with radiusr. This semi-circle rotated about its axis will generate a sphere. Find the volume of this sphere. Its volume is ^ GENERAL CONCLUSIONS From the study as here set forth, it is evident that a first coursein the elementary calculus is not too difficult for senior high-schoolpupils. This conclusion is based largely upon the following facts: 1. A historical survey of the natural growth of the calculus inthe development of mathematics shows that the subject has adapteditself slowly to the needs of the race, and that a similar adaptationto the needs of those pupils in the senior high schools who careto elect the subject is entirely possible; 2. An examination of those leading European textbooks onthe elementary calculus which have been written for beginnersshows that a number of the best teachers of secondary math-ematics do not consider the elementary calculus too difficult fortheir pupils; 3. The theory and the applications of the elementary calculuscan be so simplified
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