Find the length of the entire perimeter of the region inside r=15sin(theta)
but outside r=1.
I am not sure how to start the problem
Oops. The part from 0 to 0.0667 is not included, because it's all inside r=1. So, the real area is just
2∫[0.0667,π/2] 1/2 ((15sinθ)^2-1^2) dθ
Well, you could start by telling the r=1 circle that it's time to step aside and make way for the r=15sin(theta) party! The question is asking for the length of the entire perimeter of the region inside r=15sin(theta) but outside r=1. So, let's get this math party started!
To find the length of the entire perimeter, we need to integrate the length of each small element of the curve. In this case, we can set up the integral in terms of theta.
First, let's find the limits of integration for theta. Since we want the entire perimeter, we need to traverse the entire range of theta. For sin(theta), the range is from -pi/2 to pi/2, but when we multiply it by 15, it will cover the range from -15 to 15.
Now, we just need to find the length of a small element of the curve using a little geometry magic. The length of an arc element is given by arc length = r * d(theta), where d(theta) is a small change in theta. So, for our curve r=15sin(theta), the arc length is 15sin(theta) d(theta).
To find the length of the entire perimeter, we integrate the arc length from -15 to 15: ∫[from -15 to 15] 15sin(theta) d(theta).
Now, integrating that expression might sound intimidating, but don't worry! Just remember your trigonometric identities and you'll be fine. Once you find the antiderivative of 15sin(theta), substitute in the limits of integration, and voila! You'll have the length of the entire perimeter inside r=15sin(theta) but outside r=1.
Remember, have fun with this math party, and don't forget to invite all your geometric shapes!
To find the length of the entire perimeter of the region inside r=15sin(theta) but outside r=1, we need to first understand the shape of this region.
The equation r=15sin(theta) represents a polar curve. In polar coordinates, r represents the distance from the origin (0,0) to a point, and theta represents the angle that the line connecting the origin and the point makes with the positive x-axis.
The first equation, r=15sin(theta), represents a cardioid, which is a heart-shaped curve. The second equation, r=1, represents a circle with radius 1 centered at the origin.
To find the length of the entire perimeter of the region inside r=15sin(theta) but outside r=1, we need to find the points where these two curves intersect.
Let's solve the equations to find the intersection points.
To find the length of the entire perimeter of the region inside r=15sin(theta) but outside r=1, we need to calculate the length of the curves where these two equations intersect.
To start the problem, you can begin by plotting the two polar curves on a graph. The first curve, r=15sin(theta), is a cardioid (a heart-shaped curve), and the second curve, r=1, is a circle with radius 1 centered at the origin.
Once you've plotted the curves, you can visually determine the points where they intersect. These intersection points will determine the range of theta values you need to consider to find the length of the entire perimeter within r=15sin(theta) and outside r=1.
Once you have the range of theta values, you can set up an integral to find the length of the perimeter.
The length of the perimeter is given by the integral of the square root of the sum of the squares of the derivative of r with respect to theta, d(r/d(theta))^2, and r^2. The integral is taken with respect to theta over the range of theta values where the two curves intersect.
To evaluate this integral, you can use numerical methods, such as a numerical integration algorithm or software like Matlab, Mathematica, or Python with numerical integration libraries like scipy.
Alternatively, you can break down the region into smaller sections by finding the intersection points manually and setting up separate integrals for each section. Then, you can evaluate each integral individually and sum up the results.
I hope this helps you understand how to approach the problem. If you need further assistance with the calculations or any other questions, feel free to ask!
The area inside a polar curve is
∫1/2 r^2 dθ
The circles intersect where
15sinθ = 1
θ = 0.0667
So, from 0 to 0.0667, we have r = 15sinθ
From 0.0667 to π/2, we have to subtract the area inside r=1 from r=15sinθ, so the area is twice the area given below (due to symmetry)
∫[0,0.0667] 1/2 (15sinθ)^2 dθ
+ ∫[0.0667,π/2] 1/2 ((15sinθ)^2-1^2) dθ
or,
2(∫[0,π/2] 1/2 (15sinθ)^2 dθ - ∫[0.0667,π/2] 1/2 dθ)
Hint: the area inside 15sinθ is just the area of a circle of radius 15/2.