Application of Integrals

The area (in sq. units) enclosed between the curves y = x² and y = $\left|\mathrm{x}\right|$ is

• $\frac{2}{3}$

• $\frac{1}{6}$

• $\frac{1}{3}$

• 1

If one of the diameters of the circle, given by the equation, x²+y²−4x+6y−12=0, is a chord of a circle S, whose centre is at (−3, 2), then the radius of S is:

• 5√2

• 5√3

• 5

• 10

Let P be the point on the parabola, y²=8x which is at a minimum distance from the centre C of the circle, x²+(y+6)²=1. Then the equation of the circle, passing through C and having its centre at P is:

• x²+y²−4x+8y+12=0

• x²+y²−x+4y−12=0

• x²+y²− 4 x +2y−24=0

• x²+y²− 4 x +2y−24=0

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is:

• 4/3

• 4/√3

• 2/√3

• √3

The area (in sq units) of the region described by {x,y): y² ≤ 2x and y ≥ 4x-1} is

• 7/32

• 5/64

• 15/64

• 15/64

The area (in sq units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse

• 27/4

• 18

• 27/2

• 27/2

Let O be the vertex and Q be nay point on the parabola x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1:3 then the locus of P is 

• x²= y

• y² =x

• y² =2x

• y² =2x

If =-1 and x =2 are extreme points of f(x) =α log|x| + βx² +x, then

• α = -6, β = 1/2

• α = -6, β = -1/2

• α = 2, β = -1/2

• α = 2, β = -1/2

The area of the region described by A = {(x,y): x² +y² ≤ 1 and y² ≤1-x} is

Let the population of rabbits surviving at a time t be governed by the differential equation. If p(0) = 100 then p(t) is equal to