Application of Integrals

Let be the area of the region enclosed between the curves C1, C2, and the lines x= 1 and x= 4. Then the value of - 1 e ( -2 e- 2 ) is .

If is the area of the common region that lies inside both the given ellipses, then the value of is .

Let R denote the set of all real numbers. Then the area of the region \(x,y) R R:x>0, y> 1x, 5x-4y-1>0, 4x+4y-17<0 \ is

1. 1716- e 4
2. 338- e 4
3. 578- e 4
4. 172- e 4

Let S denote the locus of the mid-points of those chords of the parabola y2=x, such that the area of the region enclosed between the parabola and the chord is 43. Let R denote the region lying in the first quadrant, enclosed by the parabola y2=x, the curve S, and the lines x=1 and x=4. Then which of the following statements is (are) TRUE?

1. (4, 3) S
2. (5, 2) S
3. Area of R is 143-2 3
4. Area of R is 143- 3

Let S= \(x,y) R R: x 0, y 0, y2 4x, y2 12-2x and 3y+ 8 x 5 8 \. If the area of the region S is 2, then is equal to

1. 172
2. 173
3. 174
4. 175