ainster31 said:Homework Statement
http://i.imgur.com/d4ViHux.png
Homework Equations
The Attempt at a Solution
The author writes: "Now, using symmetry, we have..."
But what symmetry does the author use? Also, I got the integral as shown in the remark but why is it wrong?
cepheid said:The symmetry that's used is only to integrate over only one semi-circle in the xy-plane, and not the other. That's why θ only ranges from 0 to π/2, not all the way to π. That's also why there is a factor of 2 in front of the integral: because the integral over one semi-circle should be exactly equal to the integral over the other one. This is because the portion of the hemisphere is that is above one semi-circle is equal in volume to the portion that is above the other: one portion is just the reflection of the other one across the y-axis. That is the symmetry.
You have to be careful whenever you 'execute' a square root. √(x2) is not the same as x.ainster31 said:Alright, but why is the integral shown in the remark incorrect?
Symmetry is a mathematical concept that refers to a balanced, regular pattern or arrangement. In the context of double integrals, symmetry is used to simplify the integration process by taking advantage of the symmetry of the function being integrated. This means that instead of evaluating the integral over the entire domain, it can be broken down into smaller, symmetrical regions, which can then be summed together to obtain the final result.
Using symmetry in solving double integrals is not always necessary, but it can greatly simplify the integration process and make it more efficient. By breaking down the integral into symmetrical regions, the number of calculations needed to obtain the final result is reduced, saving time and effort.
The three types of symmetry that can be used to solve double integrals are: even symmetry, odd symmetry, and periodic symmetry. Even symmetry refers to a function that is symmetric about the y-axis, while odd symmetry refers to a function that is symmetric about the origin. Periodic symmetry refers to a function that repeats itself at regular intervals, such as a sine or cosine function.
While symmetry can be a useful tool in solving double integrals, it is not always applicable. Some functions do not exhibit any kind of symmetry, and in those cases, using symmetry to solve the integral would not be possible. Additionally, even when symmetry is present, it may not always simplify the integral to a significant extent.
Yes, symmetry can also be used to simplify the integration process for single integrals and triple integrals. Just like in the case of double integrals, symmetry can help reduce the number of calculations needed and make the integration process more efficient.