r=1- 3Sin theta¸ find the area enclosed by the inner loop
To find the area enclosed by the inner loop of the curve r = 1 - 3sin(theta), we'll first need to determine the angles theta at which the curve intersects itself. This will help us identify the boundaries for the integral that calculates the area.
Let's start by graphing the curve r = 1 - 3sin(theta):
First, we need to adjust the bounds for theta. The curve r = 1 - 3sin(theta) exists for all values of theta, so let's choose an appropriate range for a single loop. The general range for one complete loop is typically between 0 and 2*pi.
Once we have graphed the curve, we can identify the points where it intersects itself. These points correspond to the boundaries for our integral.
Now, to find the exact angles at which the curve intersects itself, we need to solve the equation:
1 - 3sin(theta) = 0
Simplifying this equation, we have:
sin(theta) = 1/3
Now, using inverse trigonometric functions, we can solve for theta:
theta = arcsin(1/3)
Remember that inverse trigonometric functions have multiple solutions, so you will obtain multiple values for theta. Find the solution within the appropriate range for one complete loop (0 to 2*pi).
Once you have identified the angles theta, you'll integrate the equation for the area formula in polar coordinates:
A = (1/2) * ∫[theta1, theta2] (r^2) d(theta)
Here, theta1 and theta2 correspond to the angles at which the curve intersects itself, as we found earlier. And r is the function of theta, in this case, r = 1 - 3sin(theta).
Evaluate this integral to find the area enclosed by the inner loop.