Find the length of the entire perimeter of the region inside
r = 16sin(theta) but outside r = 4.
First define the region. Drawing a sketch chould help.
16 sin*theta = 4 when
theta = arcsin 1/4
which cporresponds to 14.48 degrees and 165.52 degrees
Your region is the area bounded by 16 sin(theta) on the outside and r = 4 on the inside. The outer curve is called a cardioid, as I recall. There is a fomula for the arc length of a cardioid at
http://en.wikipedia.org/wiki/Cardioid
The length of the circular arc is trivial. You will need to keep theta in radians.
20*(2.888912398 - .25268002551)
64π
To find the length of the entire perimeter of the region inside the curve r = 16sin(theta) but outside r = 4, we need to determine the values of theta where the two curves intersect and then calculate the length of the curves between those points.
To find the intersection points, we can set the two equations equal to each other:
16sin(theta) = 4
Dividing both sides by 4:
4sin(theta) = 1
Rearranging the equation:
sin(theta) = 1/4
Now, we can use inverse trigonometric functions to solve for theta. Taking the inverse sine of both sides:
theta = arcsin(1/4)
Using a calculator, we find that theta is approximately 0.253 radians or about 14.5 degrees.
Now that we have the values of theta where the curves intersect, we can calculate the length of the curves between these points.
The length of a polar curve is given by the formula:
Length = ∫[a, b] sqrt(r^2 + (dr/dθ)^2) dθ
where a and b are the values of theta at the endpoints.
For the inner curve r = 16sin(theta):
Length_inner = ∫[0, arcsin(1/4)] sqrt((16sin(theta))^2 + (16cos(theta))^2) dθ
For the outer curve r = 4:
Length_outer = ∫[arcsin(1/4), 2π] sqrt((4)^2 + (0)^2) dθ
We can evaluate these integrals using numerical methods or a symbolic integrator to get the lengths of the curves.
Finally, we can add the lengths of the inner and outer curves together to get the total length of the entire perimeter of the region inside r = 16sin(theta) but outside r = 4.