Proof that area of odd-petal rose curve is the same as a circle with the petal length as its diameter.

^^^ Proof that this is a circle with radius 0.5, with its origin at (0, 0.5):

表达式12: "r" equals sine left parenthesis, "n" theta , right parenthesis

定义域\theta最小值： 0

less than or equal to theta less than or equal to

定义域\theta最大值： pi

表达式13: "n" equals 3

表达式14: "t" equals 1

技术支持

Any area highlighted in orange is the derivative of the highlighted shape's area with respect to

"t"

Notice(using the

"S"

slider) how the orange area can be distributed in parts to show how the blue or red area changes with respect to

"t"

The total area of the blue area, using that

pi "r" squared

formula, is

StartFraction, pi Over 4 , EndFraction

, so that is the area of an odd-petal rose curve, as shown by this model.

(It is double,

StartFraction, pi Over 2 , EndFraction

, for even-petal rose curves.)