A Rudimentary Treatise On The Integral Calculus Classic Reprint PDF Download

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. 1852 edition. Excerpt: ...the more the number of these parallelopipeds is increased, and their length and breadth diminished, the more nearly is their sum equal to the content of the solid AC. If the limits of the sums of the contents just written be taken in rows across the page, the result is limit hx I 'f(vy)dy + 8 / f(ai2, y)dy+... + ix j f(x, y)dy If, however, the parallelopipeds had been reckoned in rows parallel to the longest side of the page, that is, parallel to ab in the diagram, the limit of the summation would be And since both results represent the same solid content, they are equal. SECTION XI. QUADRATURE OF CURVES. 118. The Integral Calculus is applied to the rectification, or determination of the lengths of curves; to the quadrature, or determination of areas of curves; the complanation of surfaces, or determination of their superficies; and the cubature of solids, or determination of their volumes or contents. 119. The methods of determining Quadratures and Cubatures are readily demonstrated by principles already laid down. Rectification and Complanation depend on geometrical theorems, hereafter given. It has been shown, Art. 19, that if x and y be the rect. angular co-ordinates of any point of a plane curve, X, Y, and x, y the co-ordinates of its extremities, the area included by it, and straight lines from its extremities parallel to the axes of x and y respectively, is given by the formula? where it is supposed that x and y are always positive and finite, and to neither is assigned more than one value corresponding to any value of the other, between the limits X, Y, x, y. ISO. Quadrature of the Circle. Let r be the radius of the circle; x, y, its coordinates at any point referred to the centre as origin of co-ordinates; then x and y are...