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 \((r, \theta, \phi)\),
where
\(r\) is the distance from origin,
\(\theta\) is the angle between the positive \(x-axis\) and the projection of position vector on \(x-y\) plane, and
\(\phi\) is the angle between the positive \(z-axis\) and the position vector. We have, \(0 \leq \phi \leq \pi\).
We can obtain cartesian coordinates of a point from its spherical coordinates as follow:
\[x=rsin\phi \cos\theta\] \[y=rsin\phi \sin \theta\] \[z=r \cos \phi\]Similarly, we can obtain spherical coordinates of a point from its cartesian coordiantes as follows:
\[r=\sqrt{x^2+y^2+z^2}\] \[\theta = tan^{-1}\left(\dfrac{y}{x}\right)\] \[\phi = cos^{-1}\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right)\]Areas
The area \(A\) between two curves \(f(x)\) and \(g(x)\) between the limits \(x=c\) and \(x=d\) is given by:
\[A=\int_{c}^{d} f(y)-g(y) d y\]Surfaces
Let surface area be denoted by \(S\). Then,
\(S=\int 2 \pi y d s\) (rotation about \(x-axis\)), where \(ds=\sqrt{1+\left(\dfrac{d y}{d x}\right)^{2}} d x\) and \(a \leq x \leq b\)
\(S=\int 2 \pi x d s\) (rotation about \(y-axis\)), where \(ds=\sqrt{1+\left(\dfrac{d x}{d y}\right)^{2}} d y\) and \(c \leq y \leq d\)
Volumes
Case 1: \(V=\int_{c}^{d} A(y) d y\), if \(c \leq y \leq d\)
Case 2: \(V=\int_{a}^{b} A(x) d x\), if \(a \leq x \leq b\)
PYQs
Surfaces
1) Find the surface area of the plane \(x+2 y+2 z=12\) cut off by \(x=0\), \(y=0\) and \(x^{2}+y^{2}=16\).
[2016, 15M]
Volumes
1) The ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) 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 \(x^{2}+\dfrac{y^{2}}{4}=z\) which is cut off by the plane \(z=9\).
[2017, 15M]
3) Compute the volume of the solid enclosed between the surfaces \(x^2+y^2=9\) and \(x^2+z^2=9\).
[2012, 20M]
4) Find the volume of the solid that lies under the paraboloid \(z=x^2+y^2\) above the \(xy-plane\) and inside the cylinder \(x^2+y^2=2x\).
[2011, 20M]
5) Obtain the volume bounded by the elliptic paraboloid given by the equations \(z=x^{2}+9 y^{2}\) and \(z=18-x^{2}-9 y^{2}\).
[2008, 20M]
6) A figure bounded by one arch of a cycloid \(x=a(t-\sin t)\), \(y=a(1-\cos t)\), \(t \in[0,2 \pi]\) 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 \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\).
[2006, 15M]
8) Evaluate \(\iiint_{v} z d v\), 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]