What is the area of the inner loop of r = 3+6sinΘ?
you need to find the angles θ which bound the inner loop.
3+6sinθ = 0 when sinθ = -1/2
So, the loop is traced out when 7π/6 <= θ <= 11π/6
So, the area inside the loop is
∫[7π/6,11π/6] 1/2 r^2 dθ
= ∫[7π/6,11π/6] 1/2 (3+6sinθ)^2 dθ = 9π-(27√3)/2
To find the area of the inner loop of the polar curve r = 3 + 6sin(Θ), you can use the concept of double integration.
First, let's determine the bounds for the integration. The inner loop of the curve occurs when the radius, r, is negative. Since we have r = 3 + 6sin(Θ), the inner loop takes place when sin(Θ) becomes negative.
To find the bounds for the angle, we set sin(Θ) < 0 and solve for Θ:
sin(Θ) < 0
Θ < 180 degrees or Θ < π radians
Thus, the bounds for the angle will be from 0 to π radians.
Next, we need to find the bounds for the radius. Since the inner loop occurs when r is negative, we set 3 + 6sin(Θ) < 0 and solve for r:
3 + 6sin(Θ) < 0
6sin(Θ) < -3
sin(Θ) < -1/2
We know that sin(Θ) is negative in the second and third quadrants, where sin(Θ) is less than 0. Using the unit circle, we find that the angle Θ lies between 210 and 330 degrees or π/6 and 11π/6 radians.
Now, we can set up the double integral to calculate the area:
A = ∫∫R r dr dΘ
Where R represents the region that we want to find the area of.
In this case, R will be the region defined by the polar curve r = 3 + 6sin(Θ) and the bounds we established earlier.
By evaluating the integral, we can determine the area of the inner loop. However, due to the complexity of the integral, it would be challenging to solve analytically. Therefore, it is best to use numerical integration methods or a computer software program to approximate the area.
Keep in mind that the area could be negative as it lies below the x-axis. Taking the absolute value of the calculated value will give you the actual area.