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Juggler123
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I need to find the area between the parabolas x=y^2 and x=2y-y^2, I know I need to use a double integral but am having difficulty finding the limits. Can anyone help please?
Finding the Area Between Parabolas: Double Integral Help Needed
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In summary, to find the area between the parabolas x=y^2 and x=2y-y^2, a double integral can be used. First, determine the values of x where the graphs intersect to find the limits of integration for the outer integral. Then, determine which graph is higher within that interval to determine the limits of integration for the inner integral. The resulting integral will give the area between the two parabolas.
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tiny-tim
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Hi Juggler123! 
(try using the X2 tag just above the Reply box)
No, for an area you'll only need a single integral …
divide the area into strips of width dy, find the area of each strip, and integrate over y.
(try using the X2 tag just above the Reply box
)Juggler123 said:
No, for an area you'll only need a single integral …
divide the area into strips of width dy, find the area of each strip, and integrate over y.
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HallsofIvy
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Well, you certainly could use a double integral to find area!
The double integral
[itex]\int_{x=a}^b \int_{y= f(x)}^{g(x)} 1 dy dx[/itex]
has a very simple first integral of f(x)- g(x) so that the second integral is
[itex]\int_{x=a}^b f(x)- g(x) dx[/itex]
and that is the integral for area between the curves that tiny-tim is referring to.
So, for your problem, first, determine for what values of x the graphs intersect. That will be your limits of integration a and b for the "outer" integral (a being the smaller, b the larger, of course). The decide which of the graphs is higher inside that interval. That will be f(x) and g(x), the limits of integration for the "inner" integral.
The double integral
[itex]\int_{x=a}^b \int_{y= f(x)}^{g(x)} 1 dy dx[/itex]
has a very simple first integral of f(x)- g(x) so that the second integral is
[itex]\int_{x=a}^b f(x)- g(x) dx[/itex]
and that is the integral for area between the curves that tiny-tim is referring to.
So, for your problem, first, determine for what values of x the graphs intersect. That will be your limits of integration a and b for the "outer" integral (a being the smaller, b the larger, of course). The decide which of the graphs is higher inside that interval. That will be f(x) and g(x), the limits of integration for the "inner" integral.
1. What is the purpose of finding the area between parabolas using a double integral?
The purpose of finding the area between parabolas is to calculate the total area enclosed by two intersecting parabolas. This can be useful in various mathematical and scientific applications, such as determining the volume of a shape or finding the work done by a force in physics.
2. How do you set up a double integral for finding the area between parabolas?
To set up a double integral for finding the area between parabolas, you will need to first draw a graph of the two parabolas and determine the limits of integration. Then, you will need to choose a suitable order of integration and set up the integral using the appropriate formula, such as the formula for finding the area between two curves.
3. What is the difference between a single integral and a double integral?
A single integral is used to find the area under a single curve, while a double integral is used to find the volume or area between two intersecting curves. In other words, a single integral is in one dimension, while a double integral is in two dimensions.
4. Can you use a double integral to find the area between non-parabolic curves?
Yes, a double integral can be used to find the area between any two curves, not just parabolas. The method for setting up the integral may vary depending on the shape of the curves, but the concept remains the same.
5. Are there any real-world applications for finding the area between parabolas using a double integral?
Yes, there are many real-world applications for this concept. For example, in engineering and architecture, finding the area between curves can be used to determine the volume of a 3D object or the surface area of a structure. In physics, it can be used to calculate the work done by a force, and in economics, it can be used to determine the area under a demand curve to find the total revenue.
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