A Treatise On The Integral Calculus Volume 1(1st Edition)
Authors:
Ralph Augustus Roberts
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ISBN: 1230101632, 978-1230101637
Book publisher: RareBooksClub.com
A Treatise On The Integral Calculus Volume 1 1st Edition Summary: This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1887 edition. Excerpt: ...= pds/2, where ds is the element PQ of the arc in Fig. 13, and ds = dr sec p, sin = p/r, where p is the angle PQP', namely, the angle which the radius vector makes with the curve. If we put p = r sin p in the above, we get 8=ijrta.njdr. (10) These formulae are of considerable use in cases in which the curve is such that p and r are connected by a simple relation. For example, let us consider the involute of the circle. Let P be a point on the involute, then the tangent PT to the circle is the normal to the curve; so that we have p2 = r'-a2, where OQ =p, OP = r, OT = a. p if the area be measured from the line OA, where A is the point where the involute meets the circle. Examples. 1. Show that the sectorial area of the curve r = a + b sin /, measured from a tangent drawn from the origin, is ab sin2 cp + V1 (/-sinp cos p). 2. Show that the sectorial area of the curve r1 = a2 + £' sin /, measured from a tangent drawn from the origin, is £J'sW40. 3. To find the area of the epicycloid. In this case, we have r + (m + 1)' gives r' = aW9+8±sin2fl, S= / ' sin29 (m--l)3 J =-7 r£j-(9-sin f l cos 9), (m--iy if the area be measured from the fixed circle. Hence, taking e = x, we find that the area between the curve and the radii vectores to two consecutive cusps is m (m + 1) xa2 (m-l)3 147. "We now proceed to consider the area of the general cubic, and shall show that in all cases it can be expressed by means of no higher transcendents than elliptic integrals. Let us take the axis of y parallel to the real asymptote which the cubic must always have, then it is shown in treatises on curves or the Differential Calculus that the cubic can be written y1 (ax + b) + y (aV + b'x + c') + o'V + V + c"x + d"-0. (11)...
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