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. 1857 Excerpt: ...the assigned limits, and we obtain 1 fir.-, x cos a sin a 2 a /' therefore S =-s I!----sin-1 f c, sin a J (2 a J and the limits of the integration are 0 and. Hence we cos a shall find 2 174. Instead of taking the element of the tangent plane at any point of a surface, so that its projection shall be the rectangle Ax Ay ...
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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. 1857 Excerpt: ...the assigned limits, and we obtain 1 fir.-, x cos a sin a 2 a /' therefore S =-s I!----sin-1 f c, sin a J (2 a J and the limits of the integration are 0 and. Hence we cos a shall find 2 174. Instead of taking the element of the tangent plane at any point of a surface, so that its projection shall be the rectangle Ax Ay, it may be in some cases more convenient to take it so that its projection shall be the polar element rA0 Ar. Thus we shall have S = I (sec 7 rdd dr. For example, suppose we require the area of the surface xy = az, which is cut off by the surface a? + y2 = c"; here An independent geometrical proof will be found in the Cambridge and Dublin Mathematical Journal, Vol. IX., and also in Carmichael's Treatise on the Calculus of Operations. It will be remembered that in this formula r = J(x2 + y2 + ), while in Art. 174 we denote J(a? +y2) by r. Approximate Values of Integrals. 176. Suppose y a function of x, and that we require I ydx. If the indefinite integral I ydx is known we can at once ascertain the required definite integral. If the indefinite integral is unknown, we may still determine approximately the value of the definite integral. This process of approximation is best illustrated by supposing y to be an ordinate of a curve so that I ydx represents a certain area. Divide c--a into n parts each equal to h and draw n--1 ordinates at equal distances between the initial and final ordinates; then the ordinates may be denoted by yv y2, yni yw Hence we may take (&+&+ +y.) as an approximate value of the required area. Or we may take h(y2+yz +y -) as an approximate value. We may obtain another approximation thus; suppose the extremities of the r'" and r + l'h ordinates joined; thus we have a trapezium, the a...
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Seller's Description:
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