Establishing Limits for x in Green's Theorem Practice Questions

[King_Silver]

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.

 

[eys_physics]

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.

 

[Mark44]

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.

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.

 

[King_Silver]

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!