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 tu u
英文原题
What is the length of the curve?
解析
英文解析
- What is the length of the curve?
Solution: ln 2
dx cos t sin tdy
By the Second Fundamental Theorem of Calculus, = − and = − . Theredt t dt tfore, the length of the curve is
√
√
( ) ( )
√
2 2
∫ ∫ ∫
2 √ 2 222
dx dy cos t sin t 1
√2 + dt = + dt = dt = [ln t ] = ln 2 .
2 21
dt dt t t t
1 1 1
∫ ∫
1 1
′ 2 ′