HMMT 二月 1998 · CALC 赛 · 第 3 题

英文解析

3. Problem: Find the area of the region bounded by the graphs of y = x , y = x , and x = 2.
   Solution: There are two regions to consider. First, there is the region bounded by y = x and y = x , in 2
   the interval [0 , 1]. In this interval, the values of y = x are greater than the values of y = x , thus the area is 2
   ∫
   ( )1
   calculated by x − x dx .2
   Second, there is the region bounded by y = x and y = x and x = 2, in the interval [1 , 2]. In this interval,20
   ∫
   ( )2
   the values of y = x are greater than the values of y = x , thus the area is calculated by x − x dx .22
   ∫ ∫1
   1 2
   ( ) ( )
   2 2
   Then the total area of the region bounded by the three graphs is x − x dx + x − x dx = 1 .
   0 1
   √
   ∫
   f ( x ) dx 1
   2 3
   x x x 0