Area of a circle "ds" over the whole circle formula_33 is just the arc length, which is its circumference, so this shows that the area "A" enclosed by the circle is equal to formula_34 times the circumference of the circle. Note that the area of a semicircle of radius r can be computed by the integral formula_35. By trigonometric substitution, we substitute formula_36, hence formula_37 The last step follows since the trigonometric identity formula_42 implies that formula_43 and formula_44 have equal integrals over the interval formula_45, using integration by substitution. But on the other hand, since formula_46, the sum of the two integrals is
The Doll that Came Straight from Fairyland a cannon and a magical balloon, but only replies that his godmother should give him what she thinks best. In response, Fairy Zigzag instead asks Princess Pansy what she would like. Princess Pansy asks for a fine clay doll. The Fairy Zigzag creates a magical doll straight from fairyland and then leaves the party. The Prince is left without a gift from his fairy godmother. Princess Pansy names the doll Lady Emmelina and in the days following the party becomes bewitched by the doll. Prince Perfection gets jealous of Lady Emmelina when his sister no longer plays with him. Missing