Find the length of the entire perimeter of the region inside

r = 11 sin (theta) but outside r = 2

Make a plot of the two curves, and see if you find a way to either simplify the answer, or establish a check for your answer. The approach depends on whether you have done line integrals in your course.

Here's a plot, which unfortunately has different scales for x and y. In any case, it gives a very good idea of what you're about to calculate.

http://img515.imageshack.us/img515/2083/1287889718.png

Hint: the red curve with r=2 is a circle.

Here's a plot with proper scaling:

http://img87.imageshack.us/img87/5163/1287889718a.png

To find the length of the entire perimeter of the region inside r = 11sin(θ) but outside r = 2, we need to determine the limits of integration for theta and then calculate the length of the curves using the arc length formula.

The first step is to find the points of intersection between the two curves r = 11sin(θ) and r = 2. To do this, we set the two equations equal to each other:

11sin(θ) = 2

Next, we solve for θ. Keep in mind that since r can be negative, we need to consider both the positive and negative values when solving for θ.

sin(θ) = 2/11

θ = sin^(-1)(2/11)

θ ≈ 0.184 and θ ≈ π - 0.184 (since sin is an odd function)

For the region of interest, we need to find the length of the curves between these two values of θ. The arc length formula in polar coordinates is given by:

L = ∫(r(θ))^2 dθ

We can split the region into two parts: from θ = 0.184 to θ = π/2 and from θ = π/2 to θ = π - 0.184.

To calculate the length of the first curve from θ = 0.184 to θ = π/2, the limits of integration for the first integral will be 0.184 to π/2.

L1 = ∫[r(θ)]^2 dθ from θ = 0.184 to π/2

To calculate the length of the second curve from θ = π/2 to θ = π - 0.184, the limits of integration for the second integral will be π/2 to π - 0.184.

L2 = ∫[r(θ)]^2 dθ from θ = π/2 to π - 0.184

Finally, to find the total length of the perimeter inside the region, we add the lengths of the two curves:

Total Length = L1 + L2

Evaluate the two integrals and sum up the lengths to find the total length of the perimeter.