Answer to question 1
The given line integral is [tex]\int\limits_C {xydx+x^2dy} \,[/tex]
We evaluate the first line integral from [tex](0,0)[/tex] to [tex](4,0)[/tex].
An equation of the straight line joining [tex](0,0)[/tex] and [tex](4,0)[/tex] in the xy plane is [tex]y=0[/tex].
[tex]\Rightarrow dy=0[/tex]
The first line integral now becomes,
[tex]l_1=\int\limits^4_0 {x(0)dx+x^2(0)} \,=0[/tex]
We evaluate the second line integral from [tex](4,0)[/tex] to [tex](4,5)[/tex].
An equation of the straight line joining [tex](4,0)[/tex] and [tex](4,5)[/tex] in the xy plane is [tex]x=4[/tex].
[tex]\Rightarrow dx=0[/tex]
The second line integral now becomes,
[tex]l_2=\int\limits^5_0 {(4)y(0)+4^2dy} \,=0[/tex]
[tex]l_2=\int\limits^5_0 {16dy} \,=80[/tex]
We now evaluate the third line integral from [tex](4,5)[/tex] to [tex](0,5)[/tex].
An equation of the straight line joining [tex](4,5)[/tex] and [tex](0,5)[/tex] in the xy plane is [tex]y=5[/tex].
[tex]\Rightarrow dy=0[/tex]
The third line integral now becomes,
[tex]l_3=\int\limits^0_4 {x(5)dx+x^2(0)} \,[/tex]
[tex]l_3=\int\limits^0_4 {5xdx} \,=-40[/tex]
We now evaluate the fourth line integral from [tex](0,5)[/tex] to [tex](0,0)[/tex].
An equation of the straight line joining [tex](0,5)[/tex] and [tex](0,0)[/tex] in the xy plane is [tex]x=0[/tex].
[tex]\Rightarrow dx=0[/tex]
The fourth line integral now becomes,
[tex]l_4=\int\limits^0_5 {(0)y(0)+0^2dy} \,=0[/tex]
We now add all the line integrals to get,
[tex]l=0+80+-40+0=40[/tex]
Answer to question 2
According to the Green's Theorem,
[tex]\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[/tex].
This implies that,
[tex]\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, \int\limits^4_0 ({2x-x }) \, dxdy[/tex]
This simplifies to
[tex]\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, \int\limits^4_0 ({x }) \, dxdy[/tex]
We evaluate the inner integral to get,
[tex]\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, ({8 }) \, dy[/tex]
We now integrate again, to obtain,
[tex]\int\limits_C {xydx+x^2dy} \,=8\times 5=40 \,[/tex]
