Question

For each t ∈ (0, 1), the surface P_t 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 a_t ∈ R be the surface area of P_t. 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

The Correct Option is A, C

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.