Application of Integrals · Question 2

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)

Let P1 : y = 4x2 and P2 : y = x2 + 27 be two parabolas. If the area of the bounded region enclosed between P1 and P2 is six times the area of the bounded region enclosed between the line y = x, > 0 and P1, then is equal to :

1. 12
2. 15
3. 8
4. 6

Let A1 be the bounded area enclosed by the curves y = x2 + 2, x + y = 8 and y-axis that lies in the first quadrant. Let A2 be the bounded area enclosed by the curves y = x2 + 2, y2 = x, x = 2, and y-axis that lies in the first quadrant. Then A1 - A2 is equal to

1. 23(2 2 + 1)
2. 23(3 2 + 1)
3. 23( 2 + 1)
4. 23(4 2 + 1)