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

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

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)

Let f( ) denote the area of the region in the first quadrant bounded by x = 0, x = 1, y2 = x and y = | x - 5| - |1 - x| + x2. Then (f(0) + f(1)) is equal to

1. 12
2. 14
3. 9
4. 7

The area of the region enclosed between the circles x2 + y2 = 4 and x2 + (y - 2)2 = 4 is:

1. 23(4 - 3 3)
2. 43(2 - 3)
3. 43(2 - 3 3)
4. 23(2 - 3 3)

Let the line x = -1 divide the area of the region \(x, y) : 1 + x2 y 3 - x \ in the ratio m : n, (m, n) = 1. Then m + n is equal to

1. 27
2. 28
3. 25
4. 26

If the area of the region \(x, y) : 1 - 2x y 4 - x2, x 0, y 0\ is , , N, ( , ) = 1, then the value of ( + ) is:

1. 73
2. 85
3. 67
4. 91

Let the area of the bounded region \(x, y) : 0 9x y2, y 3x - 6 \ be A. Then 6A is equal to .

If the area of the region bounded by the curves y = 4 - x24 and y = x-42 is equal to , then 6 equals

1. 210
2. 250
3. 240
4. 220

If the area of the region \(x, y) : 1 + x2 y \x + 7, 11 - 3x\\ is A, then 3A is equal to :

1. 50
2. 46
3. 49
4. 47

If the area of the region \(x, y) : |x - 5| y 4 x\ is A, then 3A is equal to .

The area of the region \(x, y) : |x - y| y 4 x\ is

1. 5123
2. 20483
3. 512
4. 10243

If the area of the region \(x, y) : |4 - x2 | y x2, y 4, x 0 \ is ( 80 2 - ), , N, then + is equal to .

Let the area of the region \(x, y) : 2y x2 + 3, y + |x| 3, y |x - 1|\ be A. Then 6A is equal to :

1. 14
2. 18
3. 16
4. 12

The area (in sq. units) of the region \(x, y) : 0 y 2|x| + 1, 0 y x2 + 1, |x| 3\ is

1. 323
2. 643
3. 173
4. 803

The area of the region bounded by the curves x(1 + y2) = 1 and y2 = 2x is:

1. 4 - 13
2. 2 - 13
3. 2 ( 2 - 13 )
4. 12 ( 2 - 13 )

The area of the region \(x,y):x2+4x+2 y |x+2| \ is equal to

1. 7
2. 24/5
3. 20/3
4. 5

The area of the region enclosed by the curves y=ex, y=|ex-1| and y-axis is :

1. 1+ e2
2. e2
3. 1- e2
4. 2 e2-1

Let the function, f(x)= cases-3ax2-2, & x<1\ a2+bx, & x 1 cases be differentiable for all x , where a>1, b . If the area of the region enclosed by y=f(x) and the line y=-20 is + 3, , Z, then the value of + is .

The area of the region enclosed by the curves y=x2-4x+4 and y2=16-8x is :

1. 83
2. 5
3. 8
4. 43