Area Between the Parametric Curves

In summary, to find the area between the parametric curve x(t)=cos(t), y(t)=sin^2(t) and the x-axis, the formula A=∫ₐᵇ y(t)x'(t) dt is used. The curve intersects the x-axis at x=-1 and x=1. The bounds for this problem should be t values, which can be found by setting -1 and 1 equal to either cos(t) or sin^2(t). When graphing the curve on the x,y plane, it is clear that the bounds should be t values. Therefore, the minus sign in front of A is incorrect and should be removed.
  • #1
Tchao
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  • #2
Tchao said:

Homework Statement


Find the area between the parametric curve, x(t)=cos(t), y(t)=sin^2(t) and the x-axis

Homework Equations


- A = ∫ₐᵇ y(t) x'(t) dt

The Attempt at a Solution


=http://imgur.com/a/UA48d - My work shown in the link provided without the bounds, sorry for not rotating the image and the poor quality. I am not sure exactly how you will find the bounds for this problem. When I type x(t)=cos(t) and y(t)=sin^2(t) into my calculator, the curve intersects the x-axis at x = -1, 1. Would -1, 1 be the bounds for this problem or do we set both -1 and 1 equal to either cos(t) or sin^2(t), to solve for your bounds?
Why is a minus sign in front of A?
You integrate with respect to t. The bounds should be also t values. What is t when the curve crosses the x axis? Plot the curve on the x,y plane.
 

FAQ: Area Between the Parametric Curves

1. What is the definition of "Area Between the Parametric Curves"?

The area between the parametric curves refers to the region enclosed by two or more curves that are defined by parametric equations. These equations describe the relationship between the x and y coordinates of a point on the curve in terms of a third variable, usually denoted as t.

2. How is the area between parametric curves calculated?

To calculate the area between parametric curves, we use the formula A = ∫|y(t) - x(t)| dt from t = a to t = b, where y(t) and x(t) represent the y and x coordinates of the curves, and a and b represent the start and end values of the parameter t.

3. What are some real-world applications of finding the area between parametric curves?

Finding the area between parametric curves is useful in many fields, such as physics, engineering, and economics. For example, it can be used to calculate the area under a velocity-time graph to determine the displacement of an object, or to find the profit or loss of a company over a certain time period.

4. Can the area between parametric curves be negative?

Yes, the area between parametric curves can be negative. This occurs when the curves intersect and one curve is above the other. The area between the curves is then calculated as the difference between the two curves' areas, resulting in a negative value.

5. Are there any limitations when finding the area between parametric curves?

One limitation when finding the area between parametric curves is that the curves must be continuous and have a one-to-one correspondence between the parameter t and the x and y coordinates. Additionally, the curves must not intersect or overlap, as this can lead to incorrect calculations.

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