Establishing Limits for x in Green's Theorem Practice Questions

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In summary, the conversation discusses the use of Green's Theorem in solving problems involving a finite region enclosed by a parabola and a horizontal line. The speaker shares their experience with finding the limits of x and y in equations involving parabolas and horizontal lines and asks for clarification on how to establish these limits in a different situation. They are advised to review basic algebra to better understand these concepts.
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King_Silver
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I am currently practicing questions on Green's Theorem however in some questions I have been given a finite region enclosed between a parabola and a horizontal line.

In these questions I am given 2 values of y but none of x.
In one question I was given that y = x^2 and y = 9 and was immediately able to spot that x satisfies this for the values -3 and 3 therefore x [-3,3] and y [ x^2, 9]

However I now have come across a situation where y = 3x^2 and y = 5 and I cannot seem to spot any sort of relationship between these 2. Based on previous questions I figured these values of x would satisfy both values of y given.

Can anybody tell me how these limits are established? I've integrated in terms of y and now have all x terms but do not know what limits to apply to the integral.
 
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You can do in exactly the same way:
$$y=3x^2$$, divide by 3 on both sides to get
$$y/3=x^2$$, and use the same method as for ##y=x^2##. I would also recommend a figure, in order to see more clearly the region.
 
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King_Silver said:
I am currently practicing questions on Green's Theorem however in some questions I have been given a finite region enclosed between a parabola and a horizontal line.

In these questions I am given 2 values of y but none of x.
In one question I was given that y = x^2 and y = 9 and was immediately able to spot that x satisfies this for the values -3 and 3 therefore x [-3,3] and y [ x^2, 9]
What does "y [x^2, 9]" mean?
It would be much simpler and clearer to say that if x = 3 or x = -3, then y = 9.
King_Silver said:
However I now have come across a situation where y = 3x^2 and y = 5 and I cannot seem to spot any sort of relationship between these 2. Based on previous questions I figured these values of x would satisfy both values of y given.

Can anybody tell me how these limits are established? I've integrated in terms of y and now have all x terms but do not know what limits to apply to the integral.
It seems odd to me that you are working with Green's Theorem, but are having trouble solving very elementary equations such as ##3x^2 = 5##. Some time spent reviewing elementary algebra would be very beneficial.
 
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Mark44 said:
What does "y [x^2, 9]" mean?
It would be much simpler and clearer to say that if x = 3 or x = -3, then y = 9.
It seems odd to me that you are working with Green's Theorem, but are having trouble solving very elementary equations such as ##3x^2 = 5##. Some time spent reviewing elementary algebra would be very beneficial.
Sorry that should have said the limits of y are x^2 and 9 so when we integrate with respect to y those are the limits. It was a poorly written post.

That is embarrassing, I totally overlooked the other value of y despite specifying it. Brain fart. Thanks for the help!
 

1. What is the purpose of determining the limits of x?

The limits of x refer to the values that x can approach from either side of a given point. This concept is essential in calculus and is used to analyze the behavior of functions and solve problems involving rates of change.

2. How do you find the limits of x algebraically?

To find the limits of x algebraically, you can use the limit laws and algebraic manipulation to simplify the function and then evaluate the limit. You can also use the properties of limits, such as the squeeze theorem or the sandwich theorem, to determine the limit of a function.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values of x approaching the given point from one side, whereas a two-sided limit takes into account the values approaching from both sides. This can affect the value of the limit and can also indicate whether a function has a discontinuity or a removable discontinuity at a certain point.

4. Can the limit of a function at a certain point be undefined?

Yes, the limit of a function at a certain point can be undefined if the function has a vertical asymptote or if the one-sided limits from both sides do not match. This can also occur if the function has a jump discontinuity at that point.

5. How are the limits of x used in real-life applications?

The concept of limits is used in various fields such as physics, engineering, and economics to model and predict the behavior of systems and phenomena. For example, in physics, limits are used to calculate velocity and acceleration, while in economics, they are used to determine optimal production levels. Understanding limits also helps in solving optimization problems and in developing mathematical models for real-world situations.

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