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Hi, does anyone know how to find the integral that needs to be evaluated here? I can't understand how to find it from the region
Hi, does anyone know how to find the integral that needs to be evaluated here? I can't understand how to find it from the region
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Use multivariable change of coordinates.
Let u = √(yz), v = √(xz), w = √(xy).
Then, the region of integration becomes
u, v, w in [a, b], where 0 < a < b.
Next, we compute the Jacobian.
Since yz = u^2, xz = v^2, xy = w^2.
==> (xyz)^2 = (uvw)^2 ==> xyz = uvw since we're integrating in the first octant
xyz = uvw ==> xu^2 = uvw
==> x = v w/u
Similarly, y = uw/v and z = uv/w.
So, ∂(x,y,z)/∂(u,v,w) =
|-v w/u^2...w/u...v/u|
|w/v...-uw/v^2....u/v| = 4
|v/w...u/w...-uv/w^2|
Hence, the volume equals ∫∫∫ 1 dV
= ∫(u = a to b) ∫(v = a to b) ∫(w = a to b) 4 dw dv du.
= 4(b - a)^3.
I hope this helps!
Let u = √(yz), v = √(xz), w = √(xy).
Then, the region of integration becomes
u, v, w in [a, b], where 0 < a < b.
Next, we compute the Jacobian.
Since yz = u^2, xz = v^2, xy = w^2.
==> (xyz)^2 = (uvw)^2 ==> xyz = uvw since we're integrating in the first octant
xyz = uvw ==> xu^2 = uvw
==> x = v w/u
Similarly, y = uw/v and z = uv/w.
So, ∂(x,y,z)/∂(u,v,w) =
|-v w/u^2...w/u...v/u|
|w/v...-uw/v^2....u/v| = 4
|v/w...u/w...-uv/w^2|
Hence, the volume equals ∫∫∫ 1 dV
= ∫(u = a to b) ∫(v = a to b) ∫(w = a to b) 4 dw dv du.
= 4(b - a)^3.
I hope this helps!