An Elementary Treatise on the Integral Calculus, Containing Applications to Plane Curves and Surfaces, and Also Chapters on the Calculus of Variations; With Numerous Examples
An Elementary Treatise on the Integral Calculus, Containing Applications to Plane Curves and Surfaces, and Also Chapters on the Calculus of Variations; With Numerous Examples
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. 1875 Excerpt: ...of different curves. It is also easily seen, as in (130), that if a point be supposed to move round any closed boundary, the included area is in all cases represented by r2d9, taken round the entire boundary, whatever be its form; the elementary angle dd being taken with its proper sign throughout. Again, if we ...
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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. 1875 Excerpt: ...of different curves. It is also easily seen, as in (130), that if a point be supposed to move round any closed boundary, the included area is in all cases represented by r2d9, taken round the entire boundary, whatever be its form; the elementary angle dd being taken with its proper sign throughout. Again, if we transform to rectangular axes by the relations x--r cos 9, y = r sin 9, we get Hence Lambert's Theorem.--It is r dd = xdy-ydx; and the area swept out by the radius vector is represented by the integral i (xdy-ydr)' taken between suitable limits; a result which can also be. easily arrived at geometrically. 137. Area of Elliptic Sector. of importance in Astronomy to be able to express the area AFR swept out by the focal radius vector of an ellipse. This can be arrived at by integration from the polar equation of the curve; it is, however, more easily ob-t tained geometrically. For, if the ordinate PA7" be produced to meet the auxiliary circle in Q, we have where u = z. ACQ. By aid of this result, the area of any elliptic sector can he expressed in terms of the focal distances of its extremities, and of the chord joining them. For, let QFP represent the sector, and let FP = p, FQ = p', PQ = c; then, denoting by 11 and u' the eccentric angles corresponding to Pand Q, the area of the sector QFP, by (4), is A" represented by ab I..1--la-u-e(sin u--sin u ). We proceed to show that this result can be written in the form--li-p'-(sin0-smtf')! (5) S = Ac? s1n2--s1n2 t--: 2 2.. c = 2a sin-----sin------= a (cos 0'-cos j). (b) 2 2 r Again, from the ellipse, we have p = a (1-e cos M), p' = a (1--e cos M'), .'. p + p = 2---e (cos m + cos w ) = 2a-2ae cos cos 22 b + (b d--b., ... = 2a-2- cos--cos--= 2a-a (cos d- + cos d ). (c) 2 2 Hence, a...
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Add this copy of An Elementary Treatise on the Integral Calculus, to cart. $48.05, new condition, Sold by Ria Christie Books rated 4.0 out of 5 stars, ships from Uxbridge, MIDDLESEX, UNITED KINGDOM, published 2022 by Legare Street Press.
All Editions of An Elementary Treatise on the Integral Calculus, Containing Applications to Plane Curves and Surfaces, and Also Chapters on the Calculus of Variations; With Numerous Examples
