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The length of the cycloid with parametric equation x(t) = (t + sin t), y(t) = (1  cos t) between (0, 0) and (π, 2) is:
The given curve is
x(t) = (t + sin t), y(t) = (1  cos t)
Which of the following is/are true?
1.
2.
3.
4. If f(x) ≥ φ(x) and both functions are integrable in [a, b]
is revolved about major and minor axis respectively. Then the ratio of these solids made by two revolutions is:
Case (i): When ellipse is rotated about major axis:
Take a small disc at a length of x from the centre of thickness dx. Then the volume of solid obtained by rotation will be ∫(−a to a)(Area)dx
Area of disc =πr^{2}
r can be calculated from the equation of ellipse as
x^{2}/a^{2} + r^{2}/b^{2 }= 1
⇒ r^{2 }= b^{2}(1−x^{2}/a^{2}) = b^{2}/a^{2}(a^{2}−x^{2})
∴Volume of major axis = ∫(−a to a)πa^{2}/b^{2}(a^{2}−x^{2})dx
= [πb^{2}x − πb^{2}x^{3}/3a^{2}](−a to a)
= 4/3πab^{2}
Case (ii): When ellipse is rotated about minor axis:
Following similar procedure as case (i),
r^{2 }= a^{2}/b^{2}(b^{2 }− y^{2})
In this case, the area will be integrated w.r.t dy as it is rotated about the Yaxis.
∴Volume of minor axis = ∫(−a to a)πa^{2}/b^{2}(b^{2}−y^{2})dx
= [πa^{2}y − πa^{2}y^{3}/3b^{2}](−a to a)
= 4/3πab^{2}
∴ Volume about major axis/Volume about minor axis
= b/a
...(i)
...(ii)
The surface area of the segment of a sphere of radius a and height h is given by:
Let the sphere be generated by the revolution about the x  axis of the circle
x^{2} + y^{2}=a^{2} ...(i)
Let OA =a, OC = b and OB = b + h
Hence The required surface
The moment of inertia about the axis of y of the region in the xyplane bounded by y = 4  x^{2} and the xaxis, is, proportional to
The given curve is y = 4x^{2}
Hence Required moment of inertia
...(i)
...(ii)
Given that
y = log_{e}
The intrinsic equations of the cardioids r = a (1  cos θ) and r = a(1 + cos θ) measured from the pole are:
The equations of given cardioids are
r= a (l cos θ) ...(i)
r= a (1 + cos θ) ...(ii)
Intrinsic equation for cardioid (i):
...(iii)
Consider the Assertion (A) and Reason (R) given below:
Assertion (A)
Reason (R)  sin x is continuous in any closed interval [ 0 ,t] .
The correct answer is:
Let
The perimeters of the cardioids r = a (1  cos θ) and r = a (1 + cos θ) differ by:
The curves are
r = a(1  cos θ) ...(i)
r = a(1 + cos θ) ...(ii)
The perimeter for curve (i) is
Hence s_{1}  s_{2} = 8a  8a = 0.
The length of the arc of the curve 6xy = x^{4} + 3 from x = 1 to x = 2 is:
The given curve is
6xy = x^{4} + 3
Thus Length of the arc between x
= 1 and x = 2 is:
The area of the region in the first quadrant bounded by the yaxis and curves y = sin x and y = cos x is:
The given curves are
y = sin x ...(i)
y = cos x ...(ii)
The curves intersect at a point B
Hence Required area = Area OABO
The line y = x + 1 is revolved about xaxis. The volume of solid of revolution formed by revolving the area covered by the given curve, xaxis and lines x = 0, x = 2 is:
The given curve is y = x + 1. This represents a straight line
Hence The required volume
The length of the complete cycloid x = a (θ + sin θ), y = a(1  cos θ) is given by:
The equations of the curve are
x = a(θ + sin θ), y = a(1  cos θ)
Hence The entire length of the curve
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