Mathematical Methods for Physics and Engineering - Matematica.NET

2.2 INTEGRATIONyCρ(φ + dφ)dAρ(φ)ρdφBOxFigure 2.9 Finding the area of a sector OBC defined by the curve ρ(φ) andthe radii OB, OC, at angles to the x-axis φ 1 , φ 2 respectively.dA = 1 2 ρ2 dφ, as illustrated in figure 2.9, and hence the total area between twoangles φ 1 and φ 2 is given by∫ φ21A =2 ρ2 dφ. (2.38)φ 1An immediate observation is that the area of a circle of radius a is given byA =∫ 2π012 a2 dφ = [ 12 a2 φ ] 2π= πa 2 .0◮The equation in polar coordinates of an ellipse with semi-axes a and b is1ρ = cos2 φ+ sin2 φ.2 a 2 b 2Find the area A of the ellipse.Using (2.38) and symmetry, we have∫ 2πA = 1 a 2 b 2∫ π/22 0 b 2 cos 2 φ + a 2 sin 2 φ dφ =2a2 b 2 0To evaluate this integral we write t =tanφ and use (2.35):∫ ∞A =2a 2 b 20∫1∞b 2 + a 2 t dt 2 =2b201b 2 cos 2 φ + a 2 sin 2 φ dφ.1(b/a) 2 + t 2 dt.Finally, from the list of standard integrals (see subsection 2.2.3),A =2b 2 [ 1(b/a) tan−1] ∞t=2ab(b/a)0( π2 − 0 )= πab. ◭71