The following figure shows the graph of a continuous function y = f(x) on the interval [1, 3]. The points A, B, C have coordinates (1,1), (3,2),(2,3), respectively , and the lines `L_(1) and L_(2)` are parrallel, with `L_(1)` being tangent to the curve at C. If the area under the graph of y = f(x) from x = 1 to x = 3 is 4 square units, then the area of the shaded region is

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

`"Equation of "L_(2):y-1=(2-1)/(3-1)(x-1)`
`"or "2y-2=x-1`
`2y-x=1`
`rArr" Slope of "L_(2)=1//2.`
`"Equation of "L_(1)=y-3=(1)/(2)(x-2)`
`"or "2y-x=4`
`therefore" "D(1,(5)/(2)) and E(3,(7)/(2))`
Area under f(x) = 4
`therefore" Shaded area = Area of trapezium DEFG - Area under f(x)"`
`=(1)/(2)((5)/(2)+(7)/(2))xx2-4`
`=6-4=2`