Area, Volume, Mean and Root Mean Square Values

Representation of area as a double integral

Type I: Area enclosed by plane curves expressed in Cartesian co-ordinates

Area = $∫↙{a}↖{b} dx ∫↙{f_1(x)}↖{f_2(x)} dy$

Area = $∫↙{c}↖{d} dy ∫↙{g_1(x)}↖{g_2(x)} dx$

Note:

1. The area A included by curve y = f(x), the x-axis, and the ordinates x = a and x = b is given by


Area = $∫↙{a}↖{b} y dx = ∫↙{a}↖{b} f(x) dx$

2. The area A included by the curve x = f(y), the y-axis and the abscissa y = c and y = d is given by


A = $∫↙{c}↖{d} x dy = ∫↙{c}↖{d} f(y) dy$

3. Double integration is generally not suitable for parametric equations of curves. Instead we employ the following formula for area as:

Area = ∫y dx if x-axis is the boundary of the area

Area = ∫x dy if y-axis is the boundary of the area

4. Whenever we are finding the area of a plane region always consider the symmetry.

5. Area is always to be considered as positive.

Type II: Area enclosed by plane curves expressed in polar co-ordinates

Area = $∫↙{α}↖{β} \{ ∫↙{f_1(θ)}↖{f_2(θ)} r dr \} dθ$

Note:

1. The area bounded by the curve r = f(θ) and the lines θ = α and θ = β is


Area = $1/2 ∫↙{α}↖{β} r^2 dθ$

2. For the equation of the curve in implicit form, if the loop does not lie on the x or y-axis, then it is inclined to them. In case of inclined loop, we change the equation to polar co-ordinates with x = r cosθ, y = sinθ

3. Always consider the symmetry.

Mass of a lamina

Surface density ρ of a plane lamina, mass of an elementary area dA is ρ dA and total mass of the lamina is ∫ρ dA.

In cartesian co-ordinates, if ρ = f(x, y), then mass of lamina, M =
∫∫ f(x, y) dx dy

In polar co-ordinates, if
ρ = F (r, θ), the mass of lamina, M =
∫∫ F (r, θ) r dr dθ

Both the integrals being taken over area of lamina.

Volumes of solids

Volume of elementary cuboid is dx dy dz and so volume of solid is:

Volume = $∫∫↙v∫ dx dy dz$

if ρ = f (x, y, z) is the density of the solid at the point P (x, y, z), then the mass of the solid is

Mass = $∫∫↙v∫ ρ dx dy dz$ = $∫∫↙v∫ f(x, y, z) dx dy dz$

In spherical polar system, V = ∫∫∫ $r^2$ sinθ dr dθ dϕ

In cylindrical polar system, V = ∫∫∫ ρ dρ dϕ dz

Mean and R.M.S. Values

1. Mean value:


$y_m = {∫↙{a}↖{b} y dx}/{∫↙{a}↖{b} dx} = {∫↙{a}↖{b} f(x) dx}/{∫↙{a}↖{b} dx}$

2. Mean square value of y = f(x) over (a, b) is defined as:

M.S. of y = ${∫↙{a}↖{b} y^2 dx}/{∫↙{a}↖{b} dx} = {∫↙{a}↖{b} [f(x)]^2 dx}/{∫↙{a}↖{b} dx}$

3. Mean value of Z = f(x, y):

Mean value of z = f(x, y) over an area A = $Z_m$ = ${∫∫↙{A} f(x, y) dx dy} /{∫∫↙{A} dx dy}$

4. Mean value of u = f(x, y, z) over a region of volume V = $u_m$ = ${∫∫∫ f(x, y, z) dx dy dz} /{∫∫∫ dx dy dz}$

5. Root mean square value (R.M.S. value):

R.M.S. value of y = $√{{∫↙{c}↖{c+p} y^2 dx}/{∫↙{c}↖{c+p} dx}}$