Question

A triangle $ABC$ consists of vertex points $A(0,0),B(1,0)$ and $C(0,1)$. The value of the integral $\int \int 2xdxdy$ over the triangle is

• A. $1$
• B. $\frac{1}{3}$
• C. $\frac{1}{8}$
• D. $\frac{1}{9}$

Answer 1

The correct option is B $\frac{1}{3}$

Equation of line $CB$ is:
$y=1-x$
$I=\int \int 2xdxdy$



$={\int }_0^1{\int }_0^{1-x}2xdydx$
$={\int }_0^{1}2x[y{]}_0^{1-x}dx$
$={\int }_0^{1}2x(1-x)dx$
$={[x^2-\frac{2x^3}{3}]}_0^1$
$=1-\frac{2}{3}=\frac{1}{3}$