Integral Calculus -7 - Mathematics MCQ

 The given curve is
x(t) = (t + sin t), y(t) = (1 - cos t)

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 =πr²

r can be calculated from the equation of ellipse as

x²/a²​ + r²/b² = 1

⇒ r² = b²(1−x²/a²​) = b²/a²​(a²−x²)

∴Volume of major axis​ = ∫(−a to a)​πa²/b²​(a²−x²)dx

= [πb²x − πb²x³/3a²​](−a to a)

​= 4/3​πab²

Case (ii): When ellipse is rotated about minor axis:

Following similar procedure as case (i),

r² = a²/b²​(b² − y²)

In this case, the area will be integrated w.r.t dy as it is rotated about the Y-axis.

∴Volume of minor axis​ = ∫(−a to a)​πa²/b²​(b²−y²)dx

= [πa²y − πa²y³/3b²​](−a to a)

​= 4/3​πab²

∴ Volume about major axis/Volume about minor axis

 = b/a

  ...(i)

  ...(ii)

Let the sphere be generated by the revolution about the x - axis of the circle
x² + y²=a² ...(i)
Let OA =a, OC = b and OB = b + h
Hence The required surface

The given curve is y = 4x²
Hence Required moment of inertia

   ...(i)

   ...(ii)

Given that
y = log_e

The equations of given cardioids are
r= a (l -cos θ)    ...(i)
r= a (1 + cos θ)   ...(ii) 
Intrinsic equation for cardioid (i):


     ...(iii)

Let

The curves are
r = a(1 - cos θ)     ...(i)
r = a(1 + cos θ)    ...(ii)
The perimeter for curve (i) is





Hence s₁ - s₂ = 8a - 8a = 0. 

The given curve is
6xy = x⁴ + 3

Thus Length of the arc between x
= 1 and x = 2 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 given curve is y = x + 1. This represents a straight line
Hence The required volume

The equations of the curve are
x = a(θ + sin θ), y = a(1 - cos θ)

Hence The entire length of the curve