r in 4 dimensions. The sphere is the set of points such that x 1 2 + x 2 2 + x 3 2 + x 4 2 ≤ r 2 .
Let's consider how to find the volume of the sphere with radiusLet's calculate the volume by integrating along the x 1 axis. The volume is equal to V ( r ) = ∫ − r r A ( x 1 ) d x 1 , where A ( x 1 ) denotes the area element for a given value of x 1 .
What is the area element A ( x ) and what is the corresponding volume V ( r ) of the sphere ?
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(Fun in multiple dimensions == f.i.m.d)
From previous questions of f.i.m.d we concluded derivative of volume of sphere equals surface area and derivative of area of circle equals perimeter.
by similar arguments, derivative of volume in 4d equals surface area. It means that the derivative of answer of item 6 f.i.m.d question must be equal to volume. Differentiating each option of item 6 f.i.m.d question and comparing with volume options in this question we get
surface area =2((pi)^2)r^3 and volume= 1/2 ((pi^2))r^4
Since integration of A(x)dx must equal to V(x) we conclude that A(x)=(4/3)(pi)(r^2-x^2)^(3/2)
Note that you are not doing the problems in sequence. We use this item, to justify the answer of item 6.
Also, note that there are 2 options with that give the same volume. Hence, you need to justify why the area element A ( x ) is what you claim it to be.
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It can be justified that the other option is wrong as "Integral of A(x)dx(from 0 to r) " must equal to"V(x)(at r)" this is not true for the wrong option. Although my answer is just simple reasoning. But mathematically Chung kevin solved the problem very well.
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When x 1 = x , then we have x 2 2 + x 3 2 + x 4 2 ≤ r 2 − x 2 , which is a sphere of radius r 2 − x 2 . Hence, the corresponding area element would be equal to the volume of the sphere, thus A ( x ) = 3 4 π R 3 = 3 4 π r 2 − x 2 3 .
To find the volume of the sphere, we need to integrate this. We have
V ( x ) = ∫ − r r 3 4 π r 2 − x 2 3 d x
We can use the substitution x = r cos θ , d x = r − sin θ d θ , we get
3 4 π ∫ π 0 r 3 sin 3 θ × ( − r sin θ ) d θ = 3 4 π × 8 3 π r 4 = 2 1 π 2 r 4
( I skipped ∫ sin 4 θ d θ which you can do by parts.)