HMMT 二月 2005 · CALC 赛 · 第 2 题
HMMT February 2005 — CALC Round — Problem 2
题目详情
- A plane curve is parameterized by x ( t ) = du and y ( t ) = du for 1 ≤ t ≤ t t u u
解析
- What is the length of the curve? Solution: ln 2 dy dx cos t sin t By the Second Fundamental Theorem of Calculus, = − and = − . There- dt t dt t fore, the length of the curve is √ √ ( ) ( ) √ 2 2 ∫ ∫ ∫ 2 2 2 √ 2 2 dx dy cos t sin t 1 2 √
- dt = + dt = dt = [ln t ] = ln 2 . 1 2 2 dt dt t t t 1 1 1 ∫ ∫ 1 1 ′ 2 ′