Principles of the Differential and Integral Calculus; Familiarly Illustrated, and Applied to a Variety of Useful Purposes. Designed for the Instruction of Youth
General Books, Mar 6, 2012 - 32 pages
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1836 Excerpt: ...these be reduced to decimal fractions, and a sufficient number of terms subtracted from or.5, the result will be.4783055, which is the area of Ecdf. If from this we subtract the area of the right-angled triangle Ecf, which is.2165063, the remainder, or.2617992, will be the area of the sector Fcd, which is the T of the circle; Hence, 12 times this number, or 3.14159, &c. I is the area of a circle whose radius is 1. Now, as circles are as the squares of their diameters, of this, or.78539, &c. is the area of a circle whose diameter is 1. Cor. If an ellipse be described on Ab as its transverse axis, then the transverse axis will be to the conjugate as the area of the circle to the area of the ellipse. Let an ordinate be supposed to move from A uniformly along Ab, then the variable ordinate Eg will pass over the semicircle, and the variable ordinate Eh will pass over half the ellipse. But since Ck: Cm:: Eg: Eh, that is, since the transverse is to the conjugate axis, as any ordinate of the circle to the corresponding ordinate of the ellipse, the areas passed over by these ordinates must be in the same ratio. Ex. What is the area of an ellipse whose transverse axis is 20 inches and conjugate 10 inches? SECTION VIII. Surfaces Of Solids. To find the surfaces of solids of revolution. Let a solid be generated by the revolution of H the curvilinear surface Aeb, about its axis Ab. Let a circle whose plane is at right angles to Ab begin to move from A uniformly along Ab, its centre being in Ab and I its variable circumference on the surface of the solid, then it is obvious that this variable circumference will pass over the entire surface of the solid. Now the rate of increase of the arc Ae at the point E was formerly shown to be represented by Eh, the correspond...
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