24. State the limits of the integration to be used in apply ing the formula [[[dæ dy dz to find the volume of a closed surface of the second order whose equation is ax2+by+cz2+a'yz + b'xz + c'xy = 1. 25. State between what limits the integrations in JJJdx dy dz must be performed, in order to obtain the volume contained between the conical surface whose equation is and the planes whose equations are x = z and x = 0; and find the volume by this or by any other method. 26. State between what limits the integrations must be taken in order to find the volume of the solid contained between the two surfaces cz z=ax+by; and find the volume when m = n = a = b=1. = mx2 + ny2 and 27. A cavity is just large enough to allow of the complete revolution of a circular disc of radius C, whose centre describes a circle of the same radius c, while the plane of the disc is constantly parallel to a fixed plane, and perpendicular to that of the circle in which its centre moves. Shew that the volume of the cavity is 28. The axis of a right cone coincides with the generating line of a cylinder; the diameter of both cone and cylinder is equal to the common altitude; find the surface and volume of each part into which the cone is divided by the cylinder. where a is the radius of the base of the cone or cylinder. 29. Find the volume of the cono-cuneus determined by which is contained between the planes x = 0 and x=a. 30. A conoid is generated by a straight line which passes through the axis of z and is perpendicular to it. Two sections are made by parallel planes, both planes being parallel to the axis of z. Shew that the volume of the conoid included between the planes is equal to the product of the distance of the planes into half the sum of the areas of the sections made by the planes. 165 CHAPTER IX. DIFFERENTIATION OF AN INTEGRAL WITH RESPECT TO ANY QUANTITY WHICH IT MAY INVOLVE. 211. IT is sometimes necessary to differentiate an integral with respect to some quantity which it involves; this question we shall now consider. Required the differential coefficient of f(x) dx with respect to b, supposing independent of b. Let a (x) not to contain b, and a to be u = a suppose b changed into b+ Ab, in consequence of which u becomes u+Au; thus Let Ab and Au diminish without limit; thus du db = $(b). 212. Similarly, if we differentiate u with respect to a, supposing (x) not to contain a, and b to be independent of a, we obtain 213. Suppose (x) to contain a quantity c, and let it [ $ (x) dx be required to find the differential coefficient of [" a with respect to c, supposing a and b independent of c. Instead of (x) it will be convenient to write (x, c), so that the presence of the quantity c may be more clearly indicated; denote the integral by u, thus Suppose c changed into c+Ac, in consequence of which u becomes u+ Au; thus Now by the nature of a differential coefficient we have (x, c + Ac) − p (x, c) __ dp (x, c) Δε = dc where ρ is a quantity which diminishes without limit when Ac does so. Thus we have When Ac is diminished indefinitely, the second integral vanishes; for it is not greater than (b-a) p' where p' is the greatest value p can have, and p' ultimately vanishes. Hence proceeding to the limit, we have 214. It should be noticed that the preceding article supposes that neither a nor b is infinite; if, for example, b were infinite, we could not assert that (b-a) p' would necessarily vanish in the limit. 215. We have shewn then in Art. 213 that We will point out a useful application of this Suppose that (x, c) is the function of which ·(1). equation. (x, c) is the differential coefficient with respect to x, and that x (x, c) dp (x, c) is the differential coefficient is the function of which df (b, c) dy (a, c) = x (b, c) - x (a, c) (2), ....... dc let us suppose that b does not occur in (x, c), and that a is also independent of b; then (2) may be written |