12 8 10 points consider curves f x x and g x x 2 a

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8. (10 points) Consider curves f ( x ) = x and g ( x ) = x 2 . a ) Find the area in between the curves. Solution: We first need to find the points of intersection of these two curves. If x = x 2 , then squaring both sides gives us x = x 4 . Moving everything to one side and factoring yields 0 = x 4 - x = x ( x 3 - 1) = x ( x - 1)( x 2 + x + 1) Applying the quadratic equation to x 2 + x + 1, we see that it has no real roots. So the only points of intersection between f ( x ) and g ( x ) occur at x = 0 , 1. For 0 < x < 1, squaring a number makes it smaller while taking the square root makes it bigger, so f ( x ) > g ( x ) on this interval. Therefore, the desired area is just Z 1 0 f ( x ) - g ( x ) dx = Z 1 0 x - x 2 dx = 2 3 x 3 / 2 - 1 3 x 3 1 0 = 2 3 - 1 3 = 1 3 b ) Find the volume of the resulting solid if the enclosed area is rotated about the y -axis. Solution: If we rewrite the functions as functions of y , we get 13
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f ( x ) = y = x = f ( y ) = x = y 2 g ( x ) = y = x 2 = g ( y ) = x = y where we take only the positive sign of the square root because we have already established that 0 < x < 1 is positive. The two curves intersect at (0 , 0) and (1 , 1) as established in part a ), and in this case, g ( y ) > f ( y ) on the interval 0 < y < 1, so the desired volume is given by computing the following integral: π Z 1 0 g ( y ) 2 - f ( y ) 2 dy = π Z 1 0 y - y 4 dy = π 1 2 y 2 - 1 5 y 5 1 0 = π 1 2 - 1 5 = 3 π 10 14
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9. (10 points BONUS) This is a BONUS problem. Only work on this problem after you have finished rest of the exam. Cowboy Clint wants to build a dirt road from his ranch to the highway so that he can drive to the city in the shortest amount of time. The perpendicular distance from the ranch (point A) to the highway (point B) is 4 miles, and the city (C) is located 9 miles due east from down the highway from point B. Where should Clint join the dirt road to the highway if the speed limit is 20mph on the dirt road and 55mph on the highway? That is: Find the distance from point B to point X where the dirt road should join the highway.
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