Question

Evaluate the line integral by the two following methods. xy dx + x2 dy C is counterclockwise around the rectangle with vertices (0, 0), (4, 0), (4, 5), (0, 5) (a) directly (b) using Green's Theorem

Answer 1

Answer to question 1

The given line integral is $\int\limits_C {xydx+x^2dy} \,$

We evaluate the first line integral from $(0,0)$ to $(4,0)$.

An equation of the straight line joining $(0,0)$ and $(4,0)$ in the xy plane is $y=0$.

$\Rightarrow dy=0$

The first line integral now becomes,

$l_1=\int\limits^4_0 {x(0)dx+x^2(0)} \,=0$

We evaluate the second line integral from $(4,0)$ to $(4,5)$.

An equation of the straight line joining $(4,0)$ and $(4,5)$ in the xy plane is $x=4$.

$\Rightarrow dx=0$

The second line integral now becomes,

$l_2=\int\limits^5_0 {(4)y(0)+4^2dy} \,=0$

$l_2=\int\limits^5_0 {16dy} \,=80$

We now evaluate the third line integral from $(4,5)$ to $(0,5)$.

An equation of the straight line joining $(4,5)$ and $(0,5)$ in the xy plane is $y=5$.

$\Rightarrow dy=0$

The third line integral now becomes,

$l_3=\int\limits^0_4 {x(5)dx+x^2(0)} \,$

$l_3=\int\limits^0_4 {5xdx} \,=-40$

We now evaluate the fourth line integral from $(0,5)$ to $(0,0)$.

An equation of the straight line joining $(0,5)$ and $(0,0)$ in the xy plane is $x=0$.

$\Rightarrow dx=0$

The fourth line integral now becomes,

$l_4=\int\limits^0_5 {(0)y(0)+0^2dy} \,=0$

We now add all the line integrals to get,

$l=0+80+-40+0=40$

Answer to question 2

According to the Green's Theorem,

$\int\limits_C {P(x,y)dx+Q(x,y)dy} \,=\int\limits \, \int\limits_D ({\frac{\partial Q}{\partial x}}-{\frac{\partial P}{\partial y}}) \,dA$.

This implies that,

$\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, \int\limits^4_0 ({2x-x }) \, dxdy$

This simplifies to  

$\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, \int\limits^4_0 ({x }) \, dxdy$

We evaluate the inner integral to get,

$\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, ({8 }) \, dy$

We now integrate again, to obtain,

$\int\limits_C {xydx+x^2dy} \,=8\times 5=40 \,$