Find the area of the shaded region.
r^2=sin(2θ).
To find the area of the shaded region, we need to set up the integral for the area between two curves using polar coordinates.
Given that r^2 = sin(2θ), we can rewrite it in terms of r:
r^2 = sin(2θ)
r = √[(1/2)sin(4θ)]
Now, we need to find the limits of integration, θ1 and θ2, where the shaded region is bounded by the curves.
To do this, we need to solve for the values of θ that satisfy r = 0. Since r = √[(1/2)sin(4θ)], we solve for:
√[(1/2)sin(4θ)] = 0
(1/2)sin(4θ) = 0
sin(4θ) = 0
The solutions for sin(4θ) = 0 are θ = π/8, 3π/8, 5π/8, and 7π/8. These are the values of θ where the curve intersects the x-axis (r = 0).
Thus, the limits of integration are from θ1 = π/8 to θ2 = 7π/8.
Now, the formula for finding the area of a polar region between two curves is:
A = (1/2) ∫[θ1, θ2] |(r2)^2 - (r1)^2| dθ
In this case, r2 = √[(1/2)sin(4θ)] and r1 = 0 (since the inner curve is the origin).
Therefore, the equation for the area of the shaded region is:
A = (1/2) ∫[π/8, 7π/8] |[(1/2)sin(4θ)]^2 - 0| dθ
Now, we integrate this equation to find the area of the shaded region between the two curves.
Since this integral involves trigonometric functions, the evaluation of the integral can be quite difficult. If you provide specific limits of integration, I can help you evaluate the integral and find the exact area of the shaded region.
To find the area of the shaded region, we need to integrate the equation r^2 = sin(2θ) over the given range of θ and then calculate the area using the polar coordinate system.
First, let's rearrange the equation to solve for r:
r^2 = sin(2θ)
Taking the square root of both sides:
r = √(sin(2θ))
Now we need to determine the range of θ for which we want to calculate the area of the shaded region. Let's assume the range is from θ = θ1 to θ = θ2, where θ1 and θ2 are the angles at which the shaded region begins and ends, respectively.
To calculate the area, we use the formula for the area enclosed by a polar curve:
A = ½ * ∫[θ1,θ2] r^2 dθ
Substituting r = √(sin(2θ)) into the formula, we obtain:
A = ½ * ∫[θ1,θ2] (√(sin(2θ)))^2 dθ
A = ½ * ∫[θ1,θ2] sin(2θ) dθ
Now we can integrate the expression sin(2θ) with respect to θ over the given range to find the area of the shaded region.
exploiting symmetry, we can say
a = 2∫[0,π/2] 1/2 r^2 dθ
= ∫[0,π/2] sin2θ dθ = 1