take this between the assigned limits, and we obtain 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 r▲0 Ar. Thus we shall have For example, suppose we require the area of the surface xy= az, which is cut off by the surface 2+ y2=c2; here 175. Suppose x=r sin 0 cos &, y=r sin 0 sin 4, z=r cos 0, so that r, 0, are the usual polar co-ordinates of a point in space; then we shall shew hereafter that the equation 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 = √(x2 + y2+z2), while in Art. 174 we denote √(x2 + y2) by r. Approximate Values of Integrals. 176. Suppose y a function of x, and that we require Syd.c. If the indefinite integral [yde is known we can at a 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 ordi nate of a curve so that ['yda 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 y1, Y2,................ Yn yn+ Hence we may take as an approximate value. Or we may We may obtain another approximation thus; suppose the extremities of the 7th and 7+1]th ordinates joined; thus we h 2' have a trapezium, the area of which is (y,+yr+1) The sum of all such trapeziums gives as an approximate value of the area This result is in fact half the sum of the two former results. It is obvious we may make the approximation as close as we please by sufficiently increasing n. 177. The following is another method of approximation. Let a parabola be drawn having its axis parallel to that of y; let y1, y2 y3 represent three equidistant ordinates, h the distance between y1 and Y2 and therefore also between Y2 and Y 3* Then it may be proved that the area contained between the parabola, the axis of x, and the two extreme ordinates is This will be easily shewn by a figure, as the area consists of a trapezium and a parabolic segment, and the area of the latter is known by Art. 143. Let us now suppose that n is even, so that the area we have to estimate is divided into an even number of pieces. Then assume that the area of the first two pieces is that the area of the third and fourth pieces is and so on. Thus we shall have finally as an approximate result h 3 {y1 + 2 (Y ̧ + Y1 + ...... Yn-1) + Y +1 + 4 (Y2 + Y1 ......+Yn)}• 3 2 Hence we have the following rule: add together the first ordinate, the last ordinate, twice the sum of all the other odd ordinates, and four times the sum of all the even ordinates ; then multiply the result by one-third the common distance of the ordinates. EXAMPLES. 1. If A denote the area contained between the catenary, the axis of x, the axis of y, and an ordinate at the extremity of the arc s, shew that A = CS. The arc s begins at the lowest point of the curve. is Tab. (The integration may be effected by assuming x = a cos3 p.) 3. The area of the curve y (x2+ a2) = c2 (a − x) from x= =0 4. The area of the curve y3x=4a2 (2a-x) from x=0 to x=2a is 4πа2. 5. Shew that the whole area of the curve y2 supposing it bounded on one side by the asymptote, is 4a2. (Estimate the area of the loop and the other portion separately.) 6. Find the whole area between the curve y2 (x2 + a2) = a2x2 and its asymptotes. Result. 4a2. 7. Find the whole area between the curve xy=4a3 (2a-x) and its asymptote. Result. 4Tа2. 8. Find the whole area between the curve y2 (2α − x) = x3 and its asymptote. Result. 3а". 9. Find the whole area of the curve y=x+√(a2 — x2). Result. Ta2. 10. Find the area included between the curves y2-4ax=0, x2 - 4ay = 0, 16a2 Result. 3 11. Find the whole area of the curve a*y2 + b2x2 = a3b2x2. Result.ab. 12. Find the area of the loop of the curve a2y* = x* (a2 — x2). 13. The area between the tractory, the axis of y, and the 14. Find the area of the loop of the curve y2 (a2 + x2) = x2 (a2x2). Result. (π − 2). 15. Find the area of the loop of the curve 16a1y2 = b2x2 (a2 — 2ax). 16. Find the area of the loop of the curve 2y2 (a2 + x2) = (a2 — x2)2. Result. a 3/2 log (1+√2) — 2}. 17. Find the whole area of the curve 2y2 (a2 + x2) — 4ay (a2 − x2) + (a2 — x2)2 = 0. - x=B, and from the result deduce the area of the hyperbola xya between the same limits. 20. Find the area of the ellipse whose equation is |