Areas, Surfaces and Volumes
We will cover following topics
Cartesian and Polar Coordinates in Three Dimensions
The cartesian coordinates of a point are denoted by (x,y,z).
Analogous to this, the polar coordinates in three dimensions, also known as spherical coordinates, are denoted by (r,θ,ϕ),
where
r is the distance from origin,
θ is the angle between the positive x−axis and the projection of position vector on x−y plane, and
ϕ is the angle between the positive z−axis and the position vector. We have, 0≤ϕ≤π.
We can obtain cartesian coordinates of a point from its spherical coordinates as follow:
x=rsinϕcosθ y=rsinϕsinθ z=rcosϕSimilarly, we can obtain spherical coordinates of a point from its cartesian coordiantes as follows:
r=√x2+y2+z2 θ=tan−1(yx) ϕ=cos−1(z√x2+y2+z2)Areas
The area A between two curves f(x) and g(x) between the limits x=c and x=d is given by:
A=∫dcf(y)−g(y)dySurfaces
Let surface area be denoted by S. Then,
S=∫2πyds (rotation about x−axis), where ds=√1+(dydx)2dx and a≤x≤b
S=∫2πxds (rotation about y−axis), where ds=√1+(dxdy)2dy and c≤y≤d
PYQs
Surfaces
1) Find the surface area of the plane x+2y+2z=12 cut off by x=0, y=0 and x2+y2=16.
[2016, 15M]
Volumes
1) The ellipse x2a2+y2b2=1 revolves about the x−axis. Find the volume of the solid of revolution.
[2018, 13M]
2) Find the volume of the solid above the xy−plane and directly below the portion of the elliptic paraboloid x2+y24=z which is cut off by the plane z=9.
[2017, 15M]
3) Compute the volume of the solid enclosed between the surfaces x2+y2=9 and x2+z2=9.
[2012, 20M]
4) Find the volume of the solid that lies under the paraboloid z=x2+y2 above the xy−plane and inside the cylinder x2+y2=2x.
[2011, 20M]
5) Obtain the volume bounded by the elliptic paraboloid given by the equations z=x2+9y2 and z=18−x2−9y2.
[2008, 20M]
6) A figure bounded by one arch of a cycloid x=a(t−sint), y=a(1−cost), t∈[0,2π] and the x−axis is revolved about the x−axis. Find the volume of the solid of revolution.
[2007, 12M]
7) Find the volume of the uniform ellipsoid x2a2+y2b2+z2c2=1.
[2006, 15M]
8) Evaluate ∭, where V the volume is bounded below by the cone x^{2}+y^{2}=z^{2} and above by the sphere x^{2}+y^{2}+z^{2}=1 lying on the positive side of the y-axis.
[2005, 15M]
9) Find the volume generated by revolving the area bounded by the curves \left(x^{2}+4 a^{2}\right) y=8 a^{3}, 2 y=x and x=0, about the y-axis.
[2003, 15M]
10) Find the volume of the tetrahedron formed by the four planes lx+my+nz=p, lx+my=0, my+nz=0 and nz+lx=0.
[2003, 15M]
11) Find the volume of the solid generated by revolving the cardioid r=a(1-\cos \theta) about the initial line.
[2001, 15M]