Question

For each t ∈ (0, 1), the surface Pt in $\R^3$ is defined by
P_t = \left\\{(x, y, z) : (x^2 + y^2 )z = 1, t^2 ≤ x^2 + y^2 ≤ 1\right\\}.
Let at ∈ R be the surface area of Pt. Then

• A. $a_t=\iint\limits_{t^2\le x^2 + y^2\le 1}\sqrt{1+\frac{4x^2}{(x^2+y^2)^4}+\frac{4y^2}{(x^2+y^2)^4}}dx\ dy$
• B. $a_t=\iint\limits_{t^2\le x^2 + y^2\le 1}\sqrt{1+\frac{4x^2}{(x^2+y^2)^2}+\frac{4y^2}{(x^2+y^2)^2}}dx\ dy$
• C. the limit$\lim\limits_ {t→0^+}a_t$ does NOT exist
• D. the limit$\lim\limits_ {t→0^+}a_t$ exist

Answer 1

Answer: (A), (C)

$a_t=\iint\limits_{t^2\le x^2 + y^2\le 1}\sqrt{1+\frac{4x^2}{(x^2+y^2)^4}+\frac{4y^2}{(x^2+y^2)^4}}dx\ dy$

Answer 2

The correct option is (A) : $a_t=\iint\limits_{t^2\le x^2 + y^2\le 1}\sqrt{1+\frac{4x^2}{(x^2+y^2)^4}+\frac{4y^2}{(x^2+y^2)^4}}dx\ dy$ and (C) : the limit$\lim\limits_ {t→0^+}a_t$ does NOT exist.