Q 28 - Calculus

Question

Let $T$ be triangular region with vertices $(0,0),(1,1),\text{ and }(1,-1)$

If the density at each point in $T$ is proportional to the point’s distance from the $x$-axis, find the mass of $T$ .

Step-by-Step Solution

Answer

Mass of triangular region is $m=\frac{k}{3}$.

Step 1: Given Information

We need to determine mass of triangular region.

Density at each point is proportional to point's distance from $x$ axis

$\rho (x,y)=ky$

Step 2: Calculation of Mass

Mass is symmetric about $x$ axis.

Hence, total mass is twice of mass of upper half triangle.

$\Rightarrow m=2{\int }_0^1{\int }_0^{x}kydydx \left[m={\iint }_{\Omega }\rho (x,y)dA\right]$

Solving inner integral first

$m=2k{\int }_0^1{\left[\frac{y^2}{2}\right]}_0^{x}dx$

$m=k{\int }_0^1{x}^{2}dx$

$m=k{\left[\frac{x^3}{3}\right]}_0^1$

Solving further

$m=\frac{k}{3}$