- #1
Isaiasmoioso
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I can't figure how to solve this problem. Is says ∫∫(x^2)y dA where R is the region bounded by curves y=(x^2)+x and y=(x^2)-x and y=2. I can't figure how to do the limits with that. Please help!
Integrating Areas between curves
- Thread starter Isaiasmoioso
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In summary, the problem involves finding the integral of (x^2)y over a region bounded by the curves y=(x^2)+x, y=(x^2)-x, and y=2. The region is made up of two ascending parabolas with zeroes at -1, 0 and 0, 1. The solution involves splitting the integral into two double integrals and adding them together to find the total volume.
- #2
DonAntonio
- 606
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Isaiasmoioso said:
[itex]y=x^2+x[/itex] is an ascending straight parabola with zeroes at [itex]-1\,,\,0[/itex] , and [itex]y=x^2-x[/itex] is a similar such parabola with zeroes at [itex]0\,,\,1[/itex].
Well, now draw both parabolas together, find their intersection points and...voila!
DonAntonio
- #3
Isaiasmoioso
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I was able to get all that. I even graphed it but I can't figure out how to go about writing the integral.
- #4
HallsofIvy
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Where do the two parabolas intersect? Where does y= 2 intersect the two parabolas?
It looks to me that you will want to do this in two separate integrals. First, take x from the value of x where the line y= 2 intersects [itex]y= x^2+ x[/itex] to x= 0. You should be able to see that, for each x, y goes from the parabola up to [itex]y= x^2+ x[/itex]. Then do the second integral from x= 0 to the intersection of y= 2 and [itex]y= x^2- x[/itex].
It looks to me that you will want to do this in two separate integrals. First, take x from the value of x where the line y= 2 intersects [itex]y= x^2+ x[/itex] to x= 0. You should be able to see that, for each x, y goes from the parabola up to [itex]y= x^2+ x[/itex]. Then do the second integral from x= 0 to the intersection of y= 2 and [itex]y= x^2- x[/itex].
- #5
vish22
- 34
- 1
Hey man,you should do this in 2 double integrals-
0 2
∫ ∫ f(x,y) dy dx
-1 (x(x-1))
1 2
∫∫ f(x,y) dy dx
0 (x(x+1))and then add the 2 for your total volume.
I hope this is right
0 2
∫ ∫ f(x,y) dy dx
-1 (x(x-1))
1 2
∫∫ f(x,y) dy dx
0 (x(x+1))and then add the 2 for your total volume.
I hope this is right
Last edited:
Related to Integrating Areas between curves
1. What is the purpose of integrating areas between curves?
The purpose of integrating areas between curves is to find the total area enclosed by two or more curves on a graph. This can be useful in various applications, such as calculating the volume of a solid with curved surfaces or determining the displacement of an object with changing velocity.
2. How do you find the integral of a function to calculate the area between curves?
To find the integral of a function, you can use techniques such as the Fundamental Theorem of Calculus or integration by parts. Once you have the integral, you can calculate the area between curves by finding the difference between the integrals of the upper and lower curves.
3. Can you integrate areas between curves without using calculus?
No, integrating areas between curves requires the use of calculus. This is because it involves finding the area under a curve, which is represented by an integral in calculus.
4. What is the difference between finding the area between curves and finding the area under a curve?
The area between curves involves finding the total area enclosed by two or more curves, while finding the area under a curve involves finding the area between a curve and the x-axis. Additionally, the technique used for finding the area between curves is different from finding the area under a curve.
5. How can integrating areas between curves be applied in real-world situations?
Integrating areas between curves has many real-world applications, such as determining the volume of irregularly shaped objects, calculating the amount of fluid flowing through a curved pipe, or finding the total distance traveled by an object with varying velocity. It can also be used in economics to calculate the total revenue or profit of a business with changing prices.
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