Application of Integrals

Let f : (1, ) R be a function defined as f(x)= x-1x+1. Let fi+1(x)=f (fi(x) ), i=1, 2, , 25, where f1(x)=f(x). If g(x)+f26(x)=0, x (1, ), then the area of the region bounded by the curves y=g(x), 2y=2x-3, y=0 and x=4 is :

1. 18+ e 2
2. 14+ e 2
3. 56+3 e 2
4. 56+ e 2

The area of the region \(x, y) : 0 y 6-x, . . y2 4x-3, x 0 \ is :

1. 8
2. 9
3. 12
4. 15

Let e be the base of natural logarithm and let f : \1, 2, 3, 4\ \1, e, e2, e3 \ and g : \1, e, e2, e3 \ \1, 12, 13, 14 \ be two bijective functions such that f is strictly decreasing and g is strictly increasing. If (x)= [f-1 \g-1 ( 12 ) \ ]x, then the area of the region R= \(x, y) : x2 y (x), . . 0 x 1 \ is :

1. 3- e(2)3 e(2)
2. 13 e(2)
3. 3+ e(2)
4. 3+ e(2)2+ e(3)

The area of the region \(x, y) : x2-8x y -x \ is :

1. 3436
2. 6376
3. 4376
4. 5236

The area of the region R= \(x, y) : xy 27, 1 y x2 \ is equal to :

1. 78 e 3- 523
2. 54 e 3- 523
3. 54 e 3- 263
4. 54 e 3+ 263

Let f : R R be a function such that f(x)+3f ( 2-x )= x, x R. Let the maximum value of f on R be . If the area of the region bounded by the curves g(x)=x2 and h(x)= x3, >0, is 2, then 30 3 is equal to .

The area of the region \(x, y) : y -|x|, y |x x|, y 0\ is:

1. 1+ 28
2. 2+ 24
3. 28-1
4. 4+ 22

The area of the region bounded by the curves x+3y2=0 and x+4y2=1 is equal to :

1. 13
2. 23
3. 43
4. 53

If the area of the region bounded by 16x2-9y2=144 and 8x-3y=24 is A, then 3 (A+6 e(3) ) is equal to .

The area of the region R = \(x, y) : xy 8, 1 y x2, x 0\ is

1. 23(20 e(2) + 9)
2. 13(40 e(2) + 27)
3. 13(49 e(2) - 15)
4. 23(24 e(2) - 7)