How do you prove that the total area of a four-petaled rose r=sin2x is equal to one-half the area of the circumscribed circle?
Use a polar coordinate integration to get the area inside the rose. The circumscribed circle area is, of course, pi, since rmax = 1
For the rose, the area is
S (1/2)r^2 dx
(for x integrated from 0 to 2 pi
= S (1/2)sin^2x dx
"S" denotes an integral sign.
=(1/2)[(x/2) -(1/4)(sin2x)]@x=2pi -
(1/2)[(x/2) -(1/4)(sin2x)]@x=0
= (1/2)*pi
qed
To prove that the total area of a four-petaled rose with the equation r = sin(2x) is equal to one-half the area of the circumscribed circle, we need to find the areas of both shapes and compare them.
Here are the steps to determine the area of the four-petaled rose:
1. Set up the integral: The area, A, of a polar curve can be found using the integral formula: A = (1/2) ∫[a, b] r^2 dθ, where r is the equation of the curve in terms of polar coordinates, and θ is the angle.
2. Determine the limits: For the four-petaled rose, we need to find the appropriate limits of integration. Since the rose has four petals, we can evaluate the area from 0 to π/2, which completes one petal. Then, we multiply the result by 4 to consider all the petals.
3. Calculate the integral: Plug in the equation r = sin(2x) into the integral formula and solve it using the appropriate limits. The resulting value will give you the area of the four-petaled rose.
Next, we will determine the area of the circumscribed circle:
1. The formula to calculate the area of a circle is A = πr^2, where r is the radius.
2. Since the radius of the circumscribed circle is equal to the maximum value of r for the four-petaled rose, we need to find the maximum value of r = sin(2x).
3. Differentiate r with respect to x, set it equal to zero, and solve for x to find the points where the maximum value occurs.
4. Once you have the x-values for the maximum points, substitute them into the equation r = sin(2x) to find the corresponding maximum value of r.
5. Finally, substitute the maximum value of r into the area formula A = πr^2 to find the area of the circumscribed circle.
After calculating both areas, compare the results and check if the area of the four-petaled rose is indeed equal to one-half the area of the circumscribed circle. If they are equal, you have successfully proved the statement.