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):

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

정의역 \theta 최솟값: 0

less than or equal to theta less than or equal to

정의역 \theta 최댓값: pi

수식 13: "n" equals 3

수식 14: "t" equals 1

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.)