Find the area of the region enclosed by the inner loop of the curve.
r = 5 + 10sin(θ)
Thanks for the help!
To find the area of the region enclosed by the inner loop of the curve, you can use the concept of polar coordinates. The equation of the curve given is in polar form, where r represents the distance from the origin and θ represents the angle measured from the positive x-axis.
The curve is defined as r = 5 + 10sin(θ). To find the area enclosed by the inner loop, we need to determine the limits of θ that define that particular region.
The inner loop is enclosed by the curve when the value of r is negative. We can set the equation r = 5 + 10sin(θ) to zero and solve for θ to find the limits.
5 + 10sin(θ) = 0
sin(θ) = -1/2
From the unit circle, we know that sin(θ) = -1/2 when θ = 7π/6 (or 210 degrees) and θ = 11π/6 (or 330 degrees).
So, the limits for θ that enclose the inner loop are θ = 7π/6 and θ = 11π/6.
To find the area enclosed, we need to integrate the equation for r with respect to θ within these limits.
A = ∫[θ1,θ2] [(1/2) * r^2] dθ
where A is the area, θ1 and θ2 are the limits (7π/6 to 11π/6), r is the equation of the curve (5 + 10sin(θ)), and dθ represents an infinitesimal change in θ.
Now, plug in the values and integrate:
A = ∫[7π/6,11π/6] [(1/2) * (5 + 10sin(θ))^2] dθ
This integral can be evaluated using trigonometric identities, simplifying the expression, and then evaluating the definite integral.
After evaluating the integral, you will get the area of the region enclosed by the inner loop of the curve.