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 $x - a x i s$ and the projection of position vector on $x - y$ plane, and $ϕ$ is the angle between the positive $z - a x i s$ and the position vector. We have, $0 \leq ϕ \leq π$ .

We can obtain cartesian coordinates of a point from its spherical coordinates as follow:

x = r s i n ϕ \cos θ

y = r s i n ϕ \sin θ

z = r \cos ϕ

Similarly, we can obtain spherical coordinates of a point from its cartesian coordiantes as follows:

r = \sqrt{x^{2} + y^{2} + z^{2}}

θ = t a n^{- 1} (\frac{y}{x})

ϕ = c o s^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})

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,

$S = \int 2 π y d s$ (rotation about $x - a x i s$ ), where $d s = \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x$ and $a \leq x \leq b$

$S = \int 2 π x d s$ (rotation about $y - a x i s$ ), where $d s = \sqrt{1 + {(\frac{d x}{d y})}^{2}} d y$ 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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ revolves about the $x - a x i s$ . Find the volume of the solid of revolution.

[2018, 13M]

2) Find the volume of the solid above the $x y - p l a n e$ and directly below the portion of the elliptic paraboloid $x^{2} + \frac{y^{2}}{4} = z$ 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 $x y - p l a n e$ and inside the cylinder $x^{2} + y^{2} = 2 x$ .

[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 $x = a (t - \sin t)$ , $y = a (1 - \cos t)$ , $t \in [0, 2 π]$ and the $x - a x i s$ is revolved about the $x - a x i s$ . Find the volume of the solid of revolution.

[2007, 12M]

7) Find the volume of the uniform ellipsoid $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$ .

[2006, 15M]

8) Evaluate $∭_{v} z d v$ , 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 $y - a x i s$ .

[2005, 15M]

9) Find the volume generated by revolving the area bounded by the curves $(x^{2} + 4 a^{2}) y = 8 a^{3}$ , $2 y = x$ and $x = 0$ , about the $y - a x i s$ .

[2003, 15M]

10) Find the volume of the tetrahedron formed by the four planes $l x + m y + n z = p$ , $l x + m y = 0$ , $m y + n z = 0$ and $n z + l x = 0$ .

[2003, 15M]

11) Find the volume of the solid generated by revolving the cardioid $r = a (1 - \cos θ)$ about the initial line.

[2001, 15M]

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