Area enclosed by the curve

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In summary, the problem is to use a line integral to find the plane area enclosed by the curve r = a*(cos t)^3 + b*(sin t)^3, where 0 < t < 2*pi. The chapter being studied is about Green's theorem in the Plane, and examples use x and y instead of t. The solution involves integrating the curve 'r' with respect to t and using polar coordinates to calculate the surface. Care must be taken to determine the correct limits of integration and a visual representation of the curve may be helpful.
  • #1
Niskamies
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The problem is the following:

Use a line integral to find the plane area enclosed by the curve r = a*(cos t)^3 + b*(sin t)^3, 0 < t <2*pi.

I don't really have a clue how to solve that. The chapter we are right now in is about Green's theorem in the Plane, and all the example problems use x and y instead of t.

I would be very glad i f somebody would tell how to solve this problem.
 
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  • #2
I would integrate the curve 'r' wrt t, taking the limit to be 0 to 2pi.
 
  • #3
This is easily done (I think) using polar coordinates.
If r(t) is the distance from the origin to the curve,
and if t is the polar angle around the origin,
then the surface is given by

Integral[r(t)²/2 dt,{t,tmin,tmax)]

You have to be careful to decide the limits of integration,
and therefore you need to make a correct graphic of this function to understand the shape you need to evaluate.
There may be some calucations to perform ...

Michel

Note:
On the lhs, you wrote r in bold.
Be careful, r is the distance in polar coordinates, it is not a vector and should not be written in bold.
 
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1. What is the significance of the area enclosed by a curve?

The area enclosed by a curve is an important concept in mathematics and science. It can represent physical quantities such as displacement, velocity, and acceleration, and is used to calculate values for these quantities in various applications.

2. How is the area enclosed by a curve calculated?

The area enclosed by a curve is typically calculated using integration techniques from calculus. This involves breaking the area into smaller, more manageable sections and summing them together to get an accurate approximation of the total area.

3. Can the area enclosed by a curve be negative?

Yes, the area enclosed by a curve can be negative. This occurs when the curve dips below the x-axis and the area below the curve is calculated as negative. It is important to pay attention to the direction of integration when calculating the area to ensure an accurate result.

4. What factors affect the area enclosed by a curve?

The shape and position of the curve, as well as the limits of integration, can affect the area enclosed by a curve. Additionally, the method of integration used can also impact the calculated area. It is important to consider all of these factors when working with curves and their enclosed areas.

5. How is the area enclosed by a curve used in real-world applications?

The area enclosed by a curve has many practical applications in fields such as physics, engineering, and economics. For example, it can be used to calculate the work done by a variable force, determine the displacement of an object, or find the profit or loss of a business over a given time period.

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