Finding AREA in POLAR COORDINATE

In summary, the homework statement asks for the area inside one leaf of the four-leaved rose. The attempted solution states that they need help with finding the lower and upper limits of integration. The solution is to find the interval [0,2pi] for which one "petal" of the rose curve is generated. The problem is then simple: take the polar area integral of the cosine function on that interval.
  • #1
muffintop
14
0

Homework Statement


Find the area inside one leaf of the four-leaved rose r = cos2x

Homework Equations


A = 1/2 antiderivative abr2 dx

The Attempt at a Solution


I just need help in finding the lower and upper limits of integration. But besides that, I know how to do the rest.
If my integration is right
A = 1/2 antiderivative cos2 2x
= 1/2 antiderivative (1 + cos 4x)/2
= 1/2 (1/2 x + 2 sin 4x​
 
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  • #2
It may help to visualize what is going on. The integral is adding many little wedges to give an area. Each wedge is a thin triangle, with the sharp point at the origin, distance "r" along the long edge, and a base of "dx" (a small angle) times r.

You want to add up all the wedges inside one rosette. So pick a value of "x" where "r" is zero. Let x increase from there, and r increase with it as you follow along the rosette. Keep going, until r comes back to zero again. You've now mapped out a single rosette. Your starting and ending values for x will be the bounds of where you want to add up all those wedges; and hence they are the bounds of the definite integral.

Cheers -- sylas

PS. Check your antiderivative, by getting its derivative again to see if you get what you started with.
 
  • #3
I'm sorry but I don't get it.
Can you show me, perhaps with a different example so it's not like you're doing the homework for me?
 
  • #4
First of all, I'm assuming r=cos(2theta) is on [0,2pi].

You need to find an interval of theta such that for r=0. This can be done by solving 0=cos(2theta) for theta. This will generate several answers. You will need to figure out which solutions will give you an interval of theta for which one "petal" of the rose curve is generated. Try graphing the polar equation and then find values of r for intermediate values between the various answers to the 0=cos(2theta) equation.

We'll call that interval [a,b], for which when values from a to b are input into the polar equation one complete petal is traced. Then the problem is simple: just take the polar area integral dA=.5 int_a^b r^2 dtheta of r=cos(2theta) on that interval.
 

1. How do you find the area in polar coordinates?

Finding the area in polar coordinates involves using the formula A = (1/2) * r^2 * θ, where r is the length of the radius and θ is the angle in radians.

2. Can you explain the concept of polar coordinates and how it relates to finding area?

Polar coordinates are a system used to locate points in a plane using a distance and an angle from a fixed point, called the pole. The distance represents the radius and the angle represents the rotation from a reference line. To find the area in polar coordinates, we use the radius and angle to calculate the sector area of a circle.

3. What are some common applications of finding area in polar coordinates?

Finding area in polar coordinates is commonly used in mathematics, physics, and engineering. It can be used to calculate the area of curved shapes, such as circles, spirals, and ellipses. It is also useful in calculating the area of sectors in polar graphs and in determining the area under a curve in calculus.

4. Are there any limitations to using polar coordinates for finding area?

One limitation of using polar coordinates to find area is that it can only be used for symmetric shapes. It also does not work well for shapes with sharp corners or edges. Additionally, calculating the area can be more complex compared to using Cartesian coordinates.

5. How does finding area in polar coordinates differ from finding area in Cartesian coordinates?

In Cartesian coordinates, the area of a shape is calculated by multiplying the length and width of a rectangle. In polar coordinates, the area is calculated using the radius and angle of a sector. This means that the shape being measured must be converted into a sector of a circle in order to use the formula for area in polar coordinates.

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