Okay, so I need to figure out the area of a pseudosphere with radius r=2 using the given metric. Let me start by recalling what a pseudosphere is. A pseudosphere is a surface of constant negative curvature, right? It's kind of like the 3D equivalent of a hyperbola in 2D. The metric provided here seems to be in 2D coordinates (x and y), but since we're talking about a pseudosphere, which is a 2D surface, maybe this metric is related to the intrinsic geometry of the pseudosphere.

The metric given is ds² = [32 / (4 - x² - y²)] (dx² + dy²). Hmm, that looks similar to some conformal metrics I've seen before. Conformal metrics have the form ds² = E(dx² + dy²), where E is a function of x and y. In this case, E is 32 divided by (4 - x² - y²). 

First, maybe I should analyze the coordinate system here. The denominator (4 - x² - y²) suggests that the coordinates x and y are defined within a region where x² + y² < 4. So the domain of this metric is the interior of a circle with radius 2. Wait, but the radius given in the problem is r=2. That might be a clue. Maybe the pseudosphere is parameterized in such a way that the curvature is related to this radius.

Now, the question asks for the area of the pseudosphere. I remember that the area of a pseudosphere can be calculated using its metric. For a surface with a metric, the area is the integral over the domain of the square root of the determinant of the metric tensor. Since this is a 2D metric, the determinant would be the coefficient E squared, but wait, no. Wait, the metric tensor for a conformal metric like this is diagonal with entries E and E. So the determinant is E * E - 0 = E². Then the area element is sqrt(det g) dx dy = E dx dy. So the area would be the integral over the domain of E dx dy.

But wait, let me confirm. The determinant of the metric tensor for a 2D conformal metric ds² = E(dx² + dy²) is indeed E², so the square root is E. Therefore, the area is the integral of E over the domain. So in this case, the area A would be the integral over x and y where x² + y² < 4 of [32 / (4 - x² - y²)] dx dy.

But integrating this might be tricky. Let's see. Maybe a change to polar coordinates would help. Let me try that. Let x = r cosθ, y = r sinθ. Then x² + y² = r², and dx dy becomes r dr dθ. The integral becomes:

A = ∫∫ [32 / (4 - r²)] * r dr dθ

The limits for r are from 0 to 2 (since x² + y² < 4), and θ goes from 0 to 2π. So:

A = ∫₀²π ∫₀² [32r / (4 - r²)] dr dθ

First, integrate over r. Let me compute the inner integral:

∫₀² [32r / (4 - r²)] dr

Let me make a substitution. Let u = 4 - r², then du/dr = -2r => -du/2 = r dr. When r=0, u=4. When r=2, u=4 - 4 = 0. So the integral becomes:

∫₄⁰ [32 / u] * (-du/2) = ∫₀⁴ [32 / u] * (du/2) = 16 ∫₀⁴ (1/u) du

But wait, ∫ (1/u) du from 0 to 4 is problematic because 1/u is not integrable over [0,4]. The integral diverges as u approaches 0. That suggests that the area is infinite. But the pseudosphere is a finite area surface, right? Wait, no. Wait, the pseudosphere has finite area. Let me think again.

Wait, maybe I made a mistake in the substitution. Let me check again. The original integral for the area is:

A = ∫₀²π ∫₀² [32r / (4 - r²)] dr dθ

But when r approaches 2, the denominator 4 - r² approaches 0, so the integrand goes to infinity. So near r=2, the integral becomes ∫ [32r / (4 - r²)] dr, which is similar to ∫ [constant / (a - r)] dr, leading to a logarithmic divergence. Wait, but when r approaches 2 from below, the denominator approaches 0, so the integrand behaves like 1/(4 - r²) ~ 1/( (2 - r)(2 + r) ), so near r=2, it's ~ 1/(4 - r). So the integral ∫ [32r / (4 - r²)] dr near r=2 becomes approximately ∫ [32*2 / (4 - r)] dr = 64 ∫ [1/(4 - r)] dr, which, when r approaches 2, gives a logarithm. But integrating from 0 to 2, the integral would be:

∫₀² [32r / (4 - r²)] dr = 16 ∫₀² [2r / (4 - r²)] dr

Wait, maybe I should do substitution u = 4 - r². Then du = -2r dr, so -du/2 = r dr. Then the integral becomes:

32 ∫ [r dr / (4 - r²)] = 32 ∫ (-du/2) / u = -16 ∫ du/u

But the limits when r=0, u=4; r=2, u=0. So:

-16 [∫₄⁰ (1/u) du] = -16 [ (-∞) ] = +∞

So the integral diverges to infinity. That would mean the area is infinite. But the pseudosphere is supposed to have finite area. Wait, something's wrong here. Maybe the metric given is not the standard pseudosphere metric, or perhaps I misinterpreted the problem.

Wait, let me recall. The standard pseudosphere (tractricoid) has constant negative curvature. The metric on the pseudosphere can be given in different coordinates. One common parametrization uses the Poincaré disk model, but that's for hyperbolic space, which has negative curvature. The pseudosphere, however, is a different surface. Wait, maybe the metric provided here is actually the metric for the pseudosphere.

Wait, let me check. The standard metric for the pseudosphere in polar coordinates is often given as ds² = dr² + (sinh² r) dθ², but that's for the hyperbolic plane. However, the pseudosphere is a different surface; it's a surface of revolution of a tractrix. Its metric can be written in terms of a radial coordinate and an angular coordinate, but the specific form might differ.

Alternatively, maybe this metric is conformally flat, and the determinant gives the area element. But according to the calculation, the area would be infinite, which contradicts the known finite area of the pseudosphere. So perhaps I'm missing something here.

Wait, maybe the radius given is not the same as the coordinate radius here. The pseudosphere has a radius of curvature, which is related to the Gaussian curvature. The Gaussian curvature K of a pseudosphere is negative. The standard pseudosphere has curvature K = -1. Wait, if the radius of curvature is r, then the curvature K is -1/r². Wait, but in the metric provided, the denominator is 4 - x² - y². The radius here might be 2, since 4 is (2)^2. So if the radius of curvature is 2, then the curvature K would be -1/(2)^2 = -1/4. But I thought the standard pseudosphere has curvature -1. Hmm, perhaps the given metric corresponds to a pseudosphere with curvature -1/4, which would make the radius of curvature 2. But the problem states "pseudosphere of radius r=2", so maybe that's consistent.

Wait, but even if the radius is 2, the area of the pseudosphere should be finite. Let me recall that the pseudosphere's area is 4π times the square of the radius. Wait, if the radius is r, then the area is 4πr². So if r=2, the area would be 4π*(2)^2 = 16π. But looking at the options, none of them have 16π. The options are A: 4π(x² - y²), B: +infty, C: 0, D: 4π(x² + y²). So perhaps none of these are correct, but the question is asking for the correct answer, so maybe I made a mistake in the area calculation.

Wait, but according to the calculation, the area integral diverges. But the pseudosphere has finite area. So perhaps the metric given is not the standard pseudosphere metric. Alternatively, maybe I made a mistake in the substitution.

Wait, let's go back. The area element in the metric is ds² = [32 / (4 - x² - y²)] (dx² + dy²). So the area element is sqrt(det g) dx dy = [32 / (4 - x² - y²)] dx dy. Then the area is the integral over the region x² + y² < 4 of [32 / (4 - x² - y²)] dx dy. Let me do this in polar coordinates again.

A = ∫₀²π ∫₀² [32r / (4 - r²)] dr dθ

The integral over θ is 2π. So:

A = 2π * ∫₀² [32r / (4 - r²)] dr

Let me compute the integral inside. Let u = 4 - r², then du = -2r dr => -du/2 = r dr. When r=0, u=4; r=2, u=0. So the integral becomes:

∫₀² [32r / (4 - r²)] dr = 32 * ∫₄⁰ [1/u] * (-du/2) = 32 * (1/2) ∫₀⁴ (1/u) du = 16 ∫₀⁴ (1/u) du

But ∫₀⁴ (1/u) du is divergent, as ln(u) evaluated from 0 to 4, which goes to -infty. So the area would be infinite. But that contradicts the problem statement, which mentions the pseudosphere of radius 2. Therefore, perhaps the given metric is not the standard pseudosphere metric, or perhaps the question is using a different parametrization.

Alternatively, maybe the area is calculated differently. Wait, let's think. The pseudosphere can be parameterized in terms of its radius. If the radius is r, then the Gaussian curvature K is -1/r². The area of the pseudosphere is 4πr². So if r=2, the area is 4π*(2)^2=16π. But none of the answer choices match that. The options are A: 4π(x² - y²), B: +infty, C:0, D:4π(x² + y²). So none of these are 16π. Therefore, perhaps the question is referring to something else.

Wait, maybe the area is computed with respect to a different metric or coordinate system. Alternatively, perhaps the given metric is not the standard pseudosphere metric but a related one. Let me check if there's a standard form for the pseudosphere metric.

Alternatively, maybe the question is asking for the Gaussian curvature. The Gaussian curvature K of a metric ds² = E(dx² + dy²) is given by -1/(2E) * (d²E/dx² + d²E/dy²), but wait, no. The Gaussian curvature for a conformal metric ds² = e^{2λ} (dx² + dy²) is K = -e^{-2λ} Δλ, where Δ is the Laplacian. Wait, maybe I need to compute the curvature of the given metric.

Alternatively, perhaps the problem is using a different parametrization. Let me compute the Gaussian curvature of the given metric. The given metric is conformal: ds² = [32 / (4 - x² - y²)] (dx² + dy²). So in terms of E, E = 32 / (4 - x² - y²). Then the Gaussian curvature K is given by:

K = - (1/(2E)) * [ ∂²E/∂x² + ∂²E/∂y² ]

Wait, no. Wait, for a conformal metric ds² = e^{2λ} (dx² + dy²), the Gaussian curvature is K = -e^{-2λ} Δλ, where Δ is the 2D Laplacian. So here, E = 32 / (4 - x² - y²) = 32 * (4 - x² - y²)^{-1}. So let me write E = 32 (4 - r²)^{-1}, where r² = x² + y².

So λ = (1/2) ln E = (1/2) ln [32 (4 - r²)^{-1}] = (1/2) [ln 32 - ln(4 - r²)].

Then the Laplacian of λ is:

Δλ = (1/r) ∂/∂r [ r ∂λ/∂r ]

Compute ∂λ/∂r:

∂λ/∂r = (1/2) [ 0 - ( -2r ) / (4 - r²) ) ] = (1/2) [ 2r / (4 - r²) ) ] = r / (4 - r²)

Then,

Δλ = (1/r) ∂/∂r [ r * (r / (4 - r²)) ) ] = (1/r) ∂/∂r [ r² / (4 - r²) ]

Compute the derivative inside:

d/dr [ r² / (4 - r²) ] = [2r(4 - r²) - r²*(-2r)] / (4 - r²)^2 = [8r - 2r³ + 2r³] / (4 - r²)^2 = 8r / (4 - r²)^2

So,

Δλ = (1/r) * [8r / (4 - r²)^2] = 8 / (4 - r²)^2

Then K = -e^{-2λ} Δλ. Let's compute e^{-2λ}:

e^{-2λ} = 1 / e^{2λ} = 1 / [32 (4 - r²)^{-1}]^{-1} ] Wait, wait, wait. Wait, E = e^{2λ}, so e^{2λ} = 32 / (4 - r²). Therefore, e^{-2λ} = (4 - r²) / 32.

Thus, K = - [ (4 - r²)/32 ] * [8 / (4 - r²)^2 ] = - [8 (4 - r²) ] / [32 (4 - r²)^2 ] = - [8 ] / [32 (4 - r²) ] = - [1/4] / (4 - r²) = -1 / [4(4 - r²)]

So the Gaussian curvature K is -1 / [4(4 - r²)].

Wait, but for a pseudosphere, the Gaussian curvature should be constant and negative. However, here K depends on r, which suggests that the given metric does not have constant curvature. That contradicts the idea that it's a pseudosphere. Therefore, perhaps the given metric is not the standard pseudosphere metric. Therefore, maybe the problem is not correctly stated, or perhaps I'm misunderstanding something.

Alternatively, maybe the pseudosphere in question is a different kind, perhaps a tractricoid, which has constant negative curvature. The standard tractricoid has Gaussian curvature K = -1/a², where a is the radius. If the radius is 2, then K = -1/4. But according to the calculation above, the curvature here is -1/[4(4 - r²)], which depends on r, so it's not constant. Therefore, this metric does not represent a surface of constant curvature, hence not a pseudosphere.

But the problem states that it's the pseudosphere of radius r=2. So perhaps the given metric is not the standard one, but a different parametrization. Alternatively, maybe the question is referring to the Gaussian curvature, but the options don't include any curvature expressions.

Wait, looking at the options again: A, B, C, D are all expressions involving x and y. They are not numbers, so perhaps the question is asking for the area in terms of x and y, but since the area is a number, not a function, that seems unlikely. Alternatively, maybe the options are supposed to be expressions evaluated at some point, but that doesn't make sense either.

Alternatively, perhaps the problem is not well-posed, but since the options are given, I need to choose the correct one. Let me think again.

Given that the area integral diverges, as we saw, leading to an infinite area. But the pseudosphere has finite area. Therefore, perhaps the given metric is not the correct one for a pseudosphere. Alternatively, maybe I made a mistake in interpreting the radius.

Wait, another thought. The radius of curvature for a pseudosphere is related to its Gaussian curvature. If the Gaussian curvature is K = -1/R², where R is the radius of curvature. If the problem mentions a pseudosphere of radius 2, then K = -1/4. But according to the calculation above, K = -1/[4(4 - r²)], which is not constant. Therefore, the given metric does not correspond to a pseudosphere with constant curvature. Therefore, perhaps the question is incorrect, or perhaps I'm misunderstanding something.

Alternatively, maybe the radius given in the problem is the coordinate radius, not the radius of curvature. If the coordinate radius is 2, then the area would be infinite, as we saw. But the pseudosphere has finite area. Therefore, perhaps the problem is using a different parametrization, or perhaps the answer is B) +infty, but the pseudosphere is supposed to have finite area. So this is conflicting.

Alternatively, maybe the problem is referring to the area of the coordinate domain, not the actual geometric area. But the options are all expressions involving x and y, so that doesn't fit. Alternatively, maybe the area is computed as the integral of a different function. Wait, the problem says "the area of the pseudosphere of radius r=2". So perhaps the pseudosphere in question is a compact surface embedded in some space, and the given metric is its intrinsic metric. But if the integral diverges, then the area is infinite, which would correspond to option B. But the pseudosphere is a finite area surface. Therefore, this is confusing.

Wait, perhaps the problem is referring to the coordinate area, but that's not standard. Alternatively, perhaps the given metric is not the pseudosphere metric but something else. Let me check another approach.

The given metric is ds² = [32 / (4 - x² - y²)] (dx² + dy²). Let's see if this can be transformed into the standard pseudosphere metric. The standard pseudosphere (tractricoid) can be parameterized in terms of v and θ, where v is the distance along the axis of rotation, and θ is the angle around the axis. The metric of the tractricoid is given by ds² = [ (a² sin²h v) / (a² - a² sin²h v) ] dv² + (a² - a² sin²h v) dθ². Wait, maybe not. Alternatively, the tractricoid can be parameterized using a different coordinate system.

Alternatively, perhaps the given metric can be transformed into the tractricoid metric through a coordinate change. Let me consider substituting variables.

Let me try a substitution. Let me set u = x² + y². Then the denominator 4 - u. But I don't see how that helps. Alternatively, maybe use polar coordinates. Let me try polar coordinates again.

The area integral in polar coordinates is A = ∫θ ∫r [32r / (4 - r²)] dr dθ. The integral over θ is 2π. The integral over r from 0 to 2 is problematic because as r approaches 2, the denominator approaches 0, leading to a divergence. Therefore, the area is indeed infinite. But that contradicts the fact that the pseudosphere has finite area. Therefore, perhaps the problem is referring to a different surface.

Alternatively, maybe the given metric is incorrect, or perhaps the radius is not the same as the coordinate radius. Let me think again. If the radius of the pseudosphere is 2, then the Gaussian curvature K = -1/(2)^2 = -1/4. But in the given metric, the curvature is not constant, so that can't be. Therefore, the given metric must not represent a pseudosphere. Therefore, perhaps the problem is incorrect, but since we have to choose from the given options, maybe the answer is B) +infty.

But the pseudosphere is supposed to have finite area. So perhaps the answer is B) +infty, but that contradicts the standard knowledge. Alternatively, maybe the problem is referring to the coordinate area, not the geometric area. But in that case, the area would be infinite.

Alternatively, maybe the problem is referring to a different surface where the area is indeed infinite, but the term "pseudosphere" is used incorrectly. But since the problem states "pseudosphere", which is a surface of constant negative curvature, which should have finite area, I'm confused.

Wait, maybe the problem is referring to the Beltrami-Klein model or another model where the area could be infinite, but that seems unlikely. Alternatively, perhaps the answer is B) +infty, despite the pseudosphere having finite area, but given that the calculation leads to infinity, perhaps that's the intended answer.

Alternatively, maybe the problem is about the coordinate area, not the actual area. But in that case, the options are all expressions involving x and y, which would not make sense for an area. Therefore, perhaps the problem is incorrect, but since we have to choose from the options, and the calculation leads to an infinite area, the answer would be B) +infty.

But this contradicts the standard knowledge that the pseudosphere has finite area. Therefore, perhaps there's a mistake in the problem. However, given the options, and the calculation leading to an infinite area, I think the answer is B) +infty.
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