# Area between two curves (x = cos(y) and y = cos (x))

• igorrn
In summary, the author tried to find the area of a circle using cos and arccos, but couldn't find a definite answer. He then found that the area is the same as the sum of the cosine and arccosine of the radius.
igorrn
Homework Statement
Give the area marked in the graph (graph as jpg file)
Relevant Equations
x = cos y
y = cos x
x E [0,1] and y E [0,1]
I tried this:
X = cos(y) → y = arccos(x) for x E(-1,1) and y E (0,2)
Then:
There's a point I(Xi,Yi) in which:
Cos(Xi) =Arccos(Xi)
Then I said area1 (file: A1)
A1 = ∫cosx dx definite in 0, Xi
And A2 (file:A2):
A2 = ∫cosy dy definite in 0, Yi
And the overlapping area as A3 (file: A3):
A3 = ∫Yi dx definite in 0, Xi

And total area, then, is:
A = A1 + A2 - A3

I had trouble finding the value of Xi though. The best Approach I could find is 3/4, but I had not found a method further narrow the aprroach answer. I think Xi is an irrational number, I'd want to know if it has a name and definition to it like Pi or Euler's Number to find it.
I'd want to know if there's another method to calculating this area also.

#### Attachments

• Graph.jpg
32 KB · Views: 198
• A1.jpg
40.1 KB · Views: 203
• A2.jpg
34.4 KB · Views: 209
• A3.jpg
40.9 KB · Views: 228
By symmetry, the area you want is $$2 \int_0^{X} \cos x - x \,dx$$ where $X = \cos X$. Unfortunately that can only be solved numerically.

pasmith said:
By symmetry, the area you want is $$2 \int_0^{X} \cos x - x \,dx$$ where $X = \cos X$. Unfortunately that can only be solved numerically.
Thank you very much, but isn't it?
$$\int_0^{X} \ 2cos x - X \,dx$$
where $X = \cos X$.
To discount the overlapping area?
And could you explain me what is "numerically'?
(The int should be the integral symbol. Really don't know how to use it)

Last edited by a moderator:
No. There is no overlapping area if your first .jpg is the correct area. The integral$$A = 2\int_0^p \cos x - x~dx$$is correct (here ##p## is the value where ##\cos p = p##). What he means by having to do it numerically is the fact that even though you can integrate to get $$A =2\sin(p) - p^2$$you still have to find ##p## numerically. (About 0.7390851332).

igorrn said:
Thank you very much, but isn't it?
$$int_0^{X} \ 2cos x - X \,dx$$
where $X = \cos X$.
To discount the overlapping area?
Note that ##X \ne x##. Either method will get you the same answer.

igorrn said:
(The int should be the integral symbol. Really don't know how to use it)
It's this: \int
As a definite integral, \int_{a}^{b}. Note that you don't need the braces for a limit of integration that is one character, but you do need them for two or more characters. I.e., \int_0^{2 \pi}

## 1. What is the significance of finding the area between two curves?

The area between two curves is a common problem in calculus that has a variety of real-world applications. It can be used to find the total distance traveled by an object, the volume of a solid, or the total work done by a varying force. It is an important concept in understanding the relationship between two functions and their corresponding graphs.

## 2. How do you find the area between two curves x = cos(y) and y = cos(x)?

To find the area between two curves x = cos(y) and y = cos(x), you first need to determine the points of intersection between the two curves. This can be done by setting the two equations equal to each other and solving for x or y. Once you have the points of intersection, you can use the integral formula A = ∫(f(x) - g(x)) dx to find the area between the curves.

## 3. Can you use any integration method to find the area between two curves?

Yes, you can use any integration method to find the area between two curves. However, some methods may be more efficient or simpler to use depending on the specific curves and their equations. For example, if the curves are easily integrable, you may choose to use the fundamental theorem of calculus to find the area.

## 4. Are there any special cases to consider when finding the area between two curves?

Yes, there are a few special cases to consider when finding the area between two curves. One is when the curves intersect multiple times and create enclosed regions. In this case, you would need to split the integral into multiple sections to find the total area. Another case is when one curve is above the other for the entire domain, in which case the area between the curves would be equal to the integral of the top curve minus the integral of the bottom curve.

## 5. How does the orientation of the curves affect the area between them?

The orientation of the curves can affect the area between them in terms of the sign of the integral. If the curves are oriented such that the top curve is always above the bottom curve, the integral would be positive and represent the total area between the curves. However, if the curves intersect multiple times and create enclosed regions, the integral may need to be split and the signs of the individual integrals may need to be adjusted accordingly.

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