0 (0, b) Video Example ) (a,0) EXAMPLE 2 or a + and so SOLUTION Solving the equation of the ellipse for y, we get 222= Find the area enclosed by the ellipse = 1 y=: = 1. = + 2/( -32- Because the ellipse is symmetric with respect to both axes, the total area A is four times the area in the first quadrant (see the figure). The part of the ellipse in the first quadrant is given by this function. a -(%( = 0xa dx. de. To change the sin = 1, so To evaluate this integral we substitute x = a sin 8. Then dx = limits of integration we note that when x = 0, sin 0 = 0, so 0 = 0; when x = a, a-x since 0 0 1/2. Therefore [ A = 4 = 11 11 = = = a Jo 420 2ab Also 2ab 0 + r/2 a - a sin 0 = a cos 8. */2 10 r/2 a(cos e) = a|cos 8| = a cos 8 dx cos e de (1 + cos 20) de 7/2 0 de +0-0 We have shown that the area of an ellipse with semiaxes a and b is ab. In particular, taking a = b = r, we have proved the famous formula that the area of a circle with radius ris .