Application of Integrals

If the area of the region \(x, y) : ax2 y 1x, 1 x 2, 0 < a < 1 \ is ( e 2) - 17 then the value of 7a - 3 is equal to :

1. 1
2. 0
3. 2
4. -1

The area of the region enclosed by the parabolas y = x2 - 5x and y = 7x - x2 is .

The area enclosed between the curves y = x|x| and y = x - |x| is :

1. 83
2. 23
3. 43
4. 1

One of the points of intersection of the curves y=1+3x-2x2 and y= 1x is ( 12, 2 ). Let the area of the region enclosed by these curves be 124(l 5+m)-n e(1+ 5), where l, m, n N. Then l+m+n is equal to

1. 30
2. 29
3. 31
4. 32

The area (in sq. units) of the region described by \(x, y) : y2 2x, and y 4x - 1 \ is

1. 932
2. 1112
3. 89
4. 1132

The area enclosed by the curves xy+4y=16 and x+y=6 is equal to :

1. 28-30 e2
2. 30-28 e2
3. 30-32 e2
4. 32-30 e2

Three points O(0,0), P (a,a2 ), Q (-b,b2 ), a>0, b>0, are on the parabola y=x2. Let S1 be the area of the region bounded by the line PQ and the parabola, and S2 be the area of the triangle OPQ. If the minimum value of S1S2 is mn, (m,n)=1, then m+n is equal to .

The sum of squares of all possible values of k, for which area of the region bounded by the parabolas 2y2=kx and ky2=2(y-x) is maximum, is equal to :

The area of the region \(x, y): y2 4x, x<4, xy(x-1)(x-2)(x-3)(x-4)>0, x 3 \ is

1. 323
2. 163
3. 83
4. 643

The area of the region enclosed by the parabolas y=4x-x2 and 3y=(x-4)2 is equal to :

1. 329
2. 143
3. 4
4. 6