How do I find the area between two curves with one function not always on top?

In summary, to find the area of the region bounded by the curves y=x, y=1/(x^2), and x=2, one must take the integral of the top function minus the bottom function from the points where they intersect. However, in this specific case, there is no lower boundary given, so the problem may not be well-defined. To solve it, one can treat the region as two separate regions and find their respective areas, or integrate the difference of the y-values between the curves. The result is approximately 1 or log2 in base e.
  • #1
ashleyk
22
0
Find the area of the region bounded by the curves

y=x , y=1/(x^2) , and x=2

I know after you sketch it you have to take the intergral of the top function minus the bottom function from the points that they intersect. I am stuck however because one function does not appear to be on top throughout the interval. Any help would be great...
 
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  • #2
I'm thinking that since there was no other bound (you can't assume the y-axis is a bound), I imagine their point of intersection is meant to be taken as the other bound for the function.
 
  • #3
The attached plot is the area. It is:

[tex]\iint_R dA[/tex]

Right?
 

Attachments

  • area5.JPG
    area5.JPG
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  • #4
and what about area between x=0 and x=1?
 
  • #5
cronxeh said:
and what about area between x=0 and x=1?

Ok, I see what you mean. Perhaps the problem is not well defined.
 
  • #6
The region is bounded by y= x, y= 1/x2, and x= 2.
The only difficulty with that is that there is no lower boundary. I bet the problem also gives y= 0 as a boundary. In that case:

Of course, y= x and y= 1/x2 cross at (1, 1). That is, the region is bounded above by y= x from x=0 to x= 1, then by 1/x2 from x= 1 to x= 2.

The simplest way to do this is treat it as two separate regions:
1) the region bounded by y= x, y= 0, x= 1. In fact, that's a triangle with area 1/2.

2) the region bounded by x= 1, y= 1/x2, y= 0, x= 2.
Integrate y= 1/x2= x-2 from x= 1 to x= 2.

Add the areas of the two regions.
 
  • #7
saltydog said:
The attached plot is the area. It is:

[tex]\iint_R dA[/tex]

Right?

I think this is a Calc I question, so the poster wouldn't have encountered double integrals yet.
 
  • #8
Possible answer?

I got an answer of 1? can anyone verify...??
 
  • #9
How did you get 1? In fact, can you tell me what the bottom curve of your region is?
 
  • #10
answer of 1

I got the answer of one assuming the bottom is y=0...i have to ask my teacher tomorrow for sure if that is suppose to be one of the curves..
 
  • #11
i don't know how many curves you people see, but there were (from top of my head - i plotted this in the morning) 4 different regions. you have one between x=0 and x=1 - the one where 1/x^2 reaches to infinity, area is approximately 144+, then between x=1 and x=2 there is a triangular area and a parabolic area over on top.

so this question is not very well defined. you can't really find an area 'between' curves per se- as there are interlapping curves
 
  • #12
I'm 99% sure the region is meant to be this one (in really crappy, fast sketching)...
 

Attachments

  • graph.jpg
    graph.jpg
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  • #13
the intersection of y=1/(x^2) and y=x is at x=1

so the area is the area under the top curve - area under the bottom curve.

A = [I(1 to 2)(x dx)] - [I(1 to 2)((dx)/(x^2))]

I = integral sign
 
  • #14
Yes, or course! to both Moo of Doom and IntuitioN. I didn't draw my graph clear enough!

You could also just integrate the difference of the y-values
[tex]A= \int_1^2 (x- x^{-2})dx[/tex]
(that's exactly the same as what IntuitionN gave).
 
  • #15
hey,
its obvious from a book i read u equate x and 1/x^2,getting x as 1.this ends up being the lower boundary number,and then integrate definitely the multiple of x and 1/x^2gettiung the integral of 1/x,which becomes log [x].taking the definite step, the answer is log2,in base e.
 

Related to How do I find the area between two curves with one function not always on top?

1. What is the concept of "Area between two curves"?

The concept of "Area between two curves" refers to the area enclosed by two curves on a graph. It is typically used in calculus to calculate the total area between the x-axis and the two curves. This area can be positive or negative depending on the position of the curves on the graph.

2. How is the area between two curves calculated?

To calculate the area between two curves, you need to find the points of intersection between the two curves. Then, using integration, you can find the area between these points. The formula for calculating the area is ∫[upper limit, lower limit] (top curve - bottom curve) dx.

3. What is the importance of finding the area between two curves?

Finding the area between two curves is important in many fields such as physics, engineering, and economics. It allows us to calculate the volume of irregular shapes and determine the total change in a given quantity over a specific interval.

4. Can the area between two curves be negative?

Yes, the area between two curves can be negative. This happens when the bottom curve is above the top curve in certain regions. It is important to consider the positive and negative areas separately when calculating the total area.

5. Are there any specific methods for finding the area between two curves?

There are several methods for finding the area between two curves, including the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule. These methods use different formulas and techniques to approximate the area between the curves. However, integration is the most commonly used method for accurately calculating the area.

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