1.13P - Introduction to Electrodynamics

Question

Calculate the volume integral of the function $T=z$² over the tetrahedron with comers at (0,0,0), (1,0,0), (0,1,0), and (0,0,1). 

Step-by-Step Solution

Answer

The volume integral over the surface T is $-17/60$

Define the Volume integral

$1-y-z$The volume integral for a function  $f(x,y,z)$over a volume V can be calculated as $\int \int \int$$f(x,y,z)dxdydz$. The given points are joined to make the following figure.

The figure obtained is of a tetrahedron, over which the volume integral has to be evaluated

 From the figure, it can be inferred that plane formed by tetrahedron is $x+y+z=1$ , as the points intercepts at 1 on each axis. Also, the x varies from 0 to $1-y-z$, y varies from 0 to $1-z$ and z varies from 0 to 1.

+= Compute the volume integral of the given function over tetrahedron

The integral for a function  can be calculated as,

$\int \int \int f(x,y)dxdydz$

Substitute $z$²   for $f(x,y,z)$ into the equation.

$I={\int }_0^1{\int }_{y=0}^{1-z}{\int }_{x=0}^{1-y-z}z$²  $dxdydz$

$={\int }_0^1{\int }_{y=0}^{1-z}(1-y-z)z$² $dydz$

$={\int }_0^1(y-y$² $/2-yz)$^1-z₀  $z$² $dz$

$={\int }_0^1(1-z)-(1-z)$² $/2-z(1-z))z$² $dx$

Solve further as

$I={\int }_0^1(-3z +1/2+3z$² $/2)z$² 

$={\int }_0^1\underset{2}{\underbrace{z}}$² $+\underset{2}{\underbrace{3z}}$⁴ $-3z$³   $dz$

$=(\underset{6}{\underbrace{z}}$³ $+\underset{10}{\underbrace{3z}}$⁵  $-3z$⁴ $/4)$¹₀ 

$\frac{ =-17}{60}$

Thus, the volume integral over the surface  is . $= -17/60$