165. The normal of a curve touches its evolute. Determine the whole length of the evolute of an ellipse. 166. Investigate the differential coefficient of a surface of revolution. What is the surface generated by the revolution about the axis of x of the curve whose equations are of the hyperbola referred to its focus as pole; and thence determine the position of its asymptote. 168. If x, y be rectangular coordinates of any point in a curve, and r its distance from the origin, prove that the coordinates of the corresponding point in the evolute are ́d2 (r2) đ (2) 169. Having given cx (by) = ay (cz) = bz (ax), find the maximum value of each of these expressions. 170. The area of a polygon of a given number of sides, circumscribing a given oval figure, will be the least possible when each side is bisected in the point of contact. 171. Determine the multiple point in the curve whose equation is ay3 - 2ax2y x4 = 0. Find also the points where it is parallel to the coordinate axes. 172. If z = f(x, y), where x and y are both functions of dz two other variables r and 0, express and in terms of r, dz dy' 173. Apply Lagrange's theorem to find x" from the equation x2+2ax + 1 = 0. 174. What are the analytical characteristics of a point of osculation, and of a cusp? Ex. y = a vers-1 175. Trace the curve whose equation is a 3 + √2αx - x2. (1)2 + (3) 3 = 1; determine its asymptote, points of inflexion, and the angles at which it cuts the axes. 176. Eliminate by differentiation the constant from the equation √ T= x2 + √T — y2 = a (x − y). H 1821 1822 SECTION IX. . QUESTIONS IN INTEGRAL CALCULUS. 1. FIND the area of a curve in which the abscissa 0, and between the values of = 0, and 0=90°. dx axx3 dx. dx ʼn being an even number; x2√a + bx + cx2 5. How much of the Earth's surface may be seen by a person raised ʼn radii above it? 7. Find the area included between any two radii of a spiral, where the angle contained between them is the measure of their ratio. 8. Find the integral of xm dx (log x)" from x = 0 to x = 1. 9. Shew that the content of a sphere the content of the greatest cone that can be inscribed in it :: 33: 23. 10. The equation to a curve is y determine the values of the abscissa = 9x2+24x + 16; when the ordinate is a maximum, and when a minimum; and find the area included between those ordinates. 15. If there be taken the evolute of a logarithmic spiral, the evolute of that evolute, and so on ad infinitum, find the sum of the arcs of all the successive evolutes. 1823 17. Find the integrals of xdx and of v2dx, where v = log x. 18. Trace the curve whose equation is y = sec x, draw a tangent to it, and find its area. 21. The sum of the squares of the coefficients of the expan 23. In a parabola find the area included between the curve, its evolute, and its radius of curvature. 26. ACB is a quadrant of a circle whose centre is C, CA, CB its radii, AD, BE equal arcs, DE the chord of the arc DE; shew that the solid generated by the revolution of the circular segment DE, about either radius is equal to twice the sphere whose diameter is √2 sin DE. |