Integral Calculus -3 - Mathematics MCQ

We have


So, 
implies

Moment of Inertia of a Hollow Sphere about the Diameter

Suppose the mass of a hollow sphere is M, ρ is the density, inner radius R₂ and outer radius R₁, 

 ∴ M = 4/3π(R₁³ − R₂³)ρ

 Moment of inertia of a hollow sphere (I) = Moment of inertia of a solid sphere of radius R₁ - Moment of inertia of a solid sphere of radius R₂ 

Given curves are
y = e^x
y = e^-x 
From eqⁿ (i) and (ii), we get

Given curve is p = r sin α   ....(i)
Hence  



or, 

Required surface area is

The curve is r = 2 cos θ
Changing it into Cartesian coordinate, we get 
x² + y² = 2x
or, (x - l)² + y² = 1,
which is a circle with centre (1,0) and radius 1.
Hence
Perimeter = 2πr = 2π · 1 = 2π

Given curve is x² = 4ay
Differentiating, w.r. to y, we get

implies 
implies 
Hence Length of the arc between vertex and extremity of latus rectum

Given curve is y = x³

The required area = Area ABCD =

Given curve is y = log sec x
So, 
Hence Required length is given by