An Elementary Treatise On the Integral Calculus Containing Applications to Plan
An Elementary Treatise On the Integral Calculus Containing Applications to Plan
Benjamin Williamson
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2 17. From the centre of an ellipse a tangent is drawn to a semicircle described on an ordinate to the axis major ; prove that the polar equation of the locus of the point of contact is a 2 * 2 b 2 + (a 2 + P) tan 2 6 and that the whole area of the locus is 2 V / « 2 -r* 2 + * 18. Apply the three methods of approximation of Art. 148 to the calculation to 6 decimal places of the definite integral I, adopting — as the common Jo 1 +a? 12 interval in each case. Ans. (1), . 693669. (2), . 693266. (3...), . 693224. The 'eal value of the integral being log 2, or . 693147, to the same number of decimal places. 1 9. Prove that the sectorial area bounded by two focal vectors r and r' of a parabola is represented by where e is the chord of the arc, and a the semiparameter of the parabola. Examples. 221 20. Show that the whole area of the inverse of the ellipse — + — = i is a* b z represented by irk* (r I H_l\ (l_P\\ V a* b*] where a, £, are the co-ordinates of the origin of inversion, and k is the radius of the circle of inversion.
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