Question
Passage: Consider the curve C_1 given by y = e^ -x for x [0, 10 ], and the curve C_2 given by y = e^ -x ( x + x) for x [0, 10 ]. Let n be the total number of points of intersection of the curves C_1 and C_2. Suppose that _1, _2, , _n [0, 10 ] are the x-coordinates of the points of intersection of the curves C_1 and C_2 such that _1 Question: Let be the area of the region enclosed between the curves C_1, C_2, and the lines x = _1 and x = _4. Then the value of - 1 _e ( - 2 e^ - 2 ) is ___________.
Step-by-step solution
To find the points of intersection of the curves C_1 and C_2, we equate their equations: e^ -x = e^ -x ( x + x) Since e^ -x 0, we have: x + x = 1 2 (x + 4 ) = 1 (x + 4 ) = 1 2 The general solution is x + 4 = 2k + 4 or x + 4 = 2k + 3 4 for any integer k. Thus, x = 2k or x = 2k + 2 . For x [0, 10 ], the first four points of intersection are: _1 = 0 _2 = 2 _3 = 2 _4 = 2 + 2 = 5 2 The area enclosed between the curves from x = _1 to x = _4 is given by: = _ 0 ^ 5 2 |e^ -x ( x + x - 1)| dx We analyze the sign of x + x - 1 in the intervals (0, 2 ), ( 2 , 2 ), and (2 , 5 2 ): For x (0, 2 ), x + x > 1 For x ( 2 , 2 ), x + x For x (2 , 5 2 ), x + x > 1 Let I = e^ -x ( x + x - 1) dx. Using integration by parts or standard formulas, we get: e^ -x x dx = - e^ -x 2 ( x + x) e^ -x x dx = e^ -x 2 ( x - x) e^ -x ( x + x) dx = -e^ -x x Thus, I = -e^ -x x + e^ -x = e^ -x (1 - x). Now, we evaluate the area by splitting the integral: = _ 0 ^ 2 e^ -x ( x + x - 1) dx - _ 2 ^ 2 e^ -x ( x + x - 1) dx + _ 2 ^ 5 2 e^ -x ( x + x - 1) dx = [ e^ -x (1 - x) ]_ 0 ^ 2 - [ e^ -x (1 - x) ]_ 2 ^ 2 + [ e^ -x (1 - x) ]_ 2 ^ 5 2 = (e^ - 2 - 0) - (0 - e^ - 2 ) + (e^ - 5 2 - 0) = 2e^ - 2 + e^ - 5 2 We are required to find the value of - 1 _e ( - 2 e^ - 2 ): - 2 e^ - 2 = e^ - 5 2 - 1 _e (e^ - 5 2 ) = - 1 (- 5 2 ) = 5 2 = 2.5 Answer: 2.5