An Elementary Treatise On the Integral Calculus Containing Applications to Plan

Cover 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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