Advertisements
Advertisements
प्रश्न
Find the area of the region bounded by the curves (x -1)2 + y2 = 1 and x2 + y2 = 1, using integration.
Advertisements
उत्तर
The area bounded by the curves, (x – 1)2 + y2 = 1 and x2 + y 2 = 1, is represented by the shaded area as
On solving the equations, (x – 1)2 + y2 = 1 and x2 + y 2 = 1, we obtain the point of intersection as A`(1/2, sqrt3/2) and B(1/2, -sqrt3/2)`
It can be observed that the required area is symmetrical about x-axis.
∴ Area OBCAO = 2 × Area OCAO
We join AB, which intersects OC at M, such that AM is perpendicular to OC.
The coordinates of M are `(1/2, 0)`.
⇒ Area OCAD = Area OMAO + Area MCAM
= `[ int_0^(1/2) sqrt(1 - (x - 1)^2) dx + int_(1/2)^1 sqrt(1 - x^2 ) dx ]`
= `[ (x -1)/2 sqrt(1 - (x - 1)^2) + 1/2 sin^-1(x -1)]_0^(1/2) + [ x/2 sqrt(1 - x)^2 + 1/2 sin^-1 x]_(1/2)^1`
= `[ - 1/4 sqrt( 1 - (-1/2)^2) + 1/2 sin^-1(1/2 - 1) - 1/2 sin^-1 (-1)] + [1/2 sin^-1 - 1/4 sqrt(1 - (1/2)^2) - 1/2sin^-1 (1/2)]`
= `[- sqrt3/8 + 1/2(- pi/6) - 1/2 (- pi/2)] + [1/2(pi/2) - sqrt3/8 - 1/2(pi/6)]`
= `[-sqrt3/4 - pi/12 + pi/4 + pi/4 - pi/12 ]`
= `[-sqrt3/4 - pi/6 + pi/2]`
= `[2pi/6 - sqrt3/4]`
Therefore, required area OBCAO = `2 xx (2pi/6 - sqrt3/4 ) = (2pi/3 - sqrt3/2)` units
APPEARS IN
संबंधित प्रश्न
Form the differential equation of the family of circles touching the y-axis at the origin.
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
Which of the following differential equations has y = c1 ex + c2 e–x as the general solution?
(A) `(d^2y)/(dx^2) + y = 0`
(B) `(d^2y)/(dx^2) - y = 0`
(C) `(d^2y)/(dx^2) + 1 = 0`
(D) `(d^2y)/(dx^2) - 1 = 0`
Show that the family of curves for which `dy/dx = (x^2+y^2)/(2x^2)` is given by x2 - y2 = cx
Form the differential equation of the family of curves represented by y2 = (x − c)3.
Form the differential equation from the following primitive where constants are arbitrary:
y2 = 4ax
Form the differential equation from the following primitive where constants are arbitrary:
xy = a2
Form the differential equation from the following primitive where constants are arbitrary:
y = ax2 + bx + c
Form the differential equation of the family of curves represented by the equation (a being the parameter):
(x − a)2 + 2y2 = a2
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + y2 = a2
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x2 + (y − b)2 = 1
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax3
Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Show that y = bex + ce2x is a solution of the differential equation, \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0\]
Find one-parameter families of solution curves of the following differential equation:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]
Find one-parameter families of solution curves of the following differential equation:-
\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]
Find one-parameter families of solution curves of the following differential equation:-
\[\frac{dy}{dx} + y \cos x = e^{\sin x} \cos x\]
Find one-parameter families of solution curves of the following differential equation:-
\[e^{- y} \sec^2 y dy = dx + x dy\]
The differential equation which represents the family of curves y = eCx is
Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P(x, y) of the curve is equal to the distance of P from the x-axis.
The solution of the differential equation `2x * "dy"/"dx" y` = 3 represents a family of ______.
Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.
Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point (x, y) is `(y - 1)/(x^2 + x)`
Family y = Ax + A3 of curves will correspond to a differential equation of order ______.
Form the differential equation of family of circles having centre on y-axis and raduis 3 units
