Area between Curves: Calculus Solution

In summary, a question was asked about how to solve an area problem without any equations, and someone suggested using a paper cutout method. The context of the question was to find the area of region D. Another person suggested using integration, but reminded that there were no given equations. The conversation then turned to discussing Green's theorem and the possibility of it being a path integral.
  • #1
Sirsh
267
10
Hello all.

This is not homework, i stumbled upon it looking for information for my physics.. I have some background in calculus and am just wondering if anyone would have any idea on how you'd solve this area problem.. all i have is a picture of it and not any equations of the lines.

Thanks.

429px-Green%27s-theorem-simple-region.svg.png
 
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  • #2
Sirsh said:
Hello all.

This is not homework, i stumbled upon it looking for information for my physics.. I have some background in calculus and am just wondering if anyone would have any idea on how you'd solve this area problem.. all i have is a picture of it and not any equations of the lines.

Thanks.

429px-Green%27s-theorem-simple-region.svg.png

Sure. Print it, cut out the area, weigh the paper cutout (and weigh a full piece of paper with known dimensions). I still remember my undergrad Physics TA telling me about this technique...
 
  • #3
Oh, and what is the context of the question?
 
  • #4
Wow that's a very smart idea haha. Um, it was just finding out the way to calculate the area of the region D. Is there anyway mathmatically?

My thoughts are to take away the area of C1 from a-b from the area of C3 from a-b with integration.
 
  • #5
Sirsh said:
Wow that's a very smart idea haha. Um, it was just finding out the way to calculate the area of the region D. Is there anyway mathmatically?

My thoughts are to take away the area of C1 from a-b from the area of C3 from a-b with integration.

You said you had no equations. What do you plan to integrate?
 
  • #6
I notice that the curves that bound the region are marked with arrows. Is this some kind of a path integral? I seem to remember that the line integral along a closed path in a conservative field is zero.
 

What is the concept of "Area between Curves" in calculus?

The "Area between Curves" refers to the region enclosed by two or more curves on a graph in the coordinate plane. In calculus, this concept is used to find the total area under a curve or between two curves, which can be represented by an integral.

How do you find the area between two curves using calculus?

To find the area between two curves using calculus, you first need to find the points of intersection between the two curves. Then, you can set up an integral with the upper and lower limits being the x-values of these points of intersection. Finally, you can solve the integral to find the area between the curves.

What is the difference between finding the area under a curve and between two curves?

Finding the area under a curve involves finding the area between the curve and the x-axis, while finding the area between two curves involves finding the area between the two curves themselves. The latter requires finding the points of intersection between the two curves and setting up an integral with appropriate limits, while the former can be done using a single integral with the curve as the upper boundary and the x-axis as the lower boundary.

Can the area between curves be negative?

Yes, the area between curves can be negative. This can happen when the lower curve is above the upper curve in certain intervals, resulting in a negative value for the area. It is important to pay attention to the orientation of the curves and the limits of integration when calculating the area between curves.

What are some real-life applications of finding the area between curves?

Finding the area between curves has many real-life applications, such as calculating the volume of irregularly shaped objects, finding the displacement or distance traveled by an object with changing velocity, and determining the work done by a variable force. It is also used in economics to find the area under a demand or supply curve, which represents the total consumer or producer surplus.

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