Answer to question 1
The given line integral is
![\int\limits_C {xydx+x^2dy} \,](https://tex.z-dn.net/?f=%5Cint%5Climits_C%20%7Bxydx%2Bx%5E2dy%7D%20%5C%2C)
We evaluate the first line integral from
![(0,0)](https://tex.z-dn.net/?f=%280%2C0%29)
to
![(4,0)](https://tex.z-dn.net/?f=%284%2C0%29)
.
An equation of the straight line joining
![(0,0)](https://tex.z-dn.net/?f=%280%2C0%29)
and
![(4,0)](https://tex.z-dn.net/?f=%284%2C0%29)
in the xy plane is
![y=0](https://tex.z-dn.net/?f=y%3D0)
.
![\Rightarrow dy=0](https://tex.z-dn.net/?f=%5CRightarrow%20dy%3D0)
The first line integral now becomes,
![l_1=\int\limits^4_0 {x(0)dx+x^2(0)} \,=0](https://tex.z-dn.net/?f=l_1%3D%5Cint%5Climits%5E4_0%20%7Bx%280%29dx%2Bx%5E2%280%29%7D%20%5C%2C%3D0)
We evaluate the second line integral from
![(4,0)](https://tex.z-dn.net/?f=%284%2C0%29)
to
![(4,5)](https://tex.z-dn.net/?f=%284%2C5%29)
.
An equation of the straight line joining
![(4,0)](https://tex.z-dn.net/?f=%284%2C0%29)
and
![(4,5)](https://tex.z-dn.net/?f=%284%2C5%29)
in the xy plane is
![x=4](https://tex.z-dn.net/?f=x%3D4)
.
![\Rightarrow dx=0](https://tex.z-dn.net/?f=%5CRightarrow%20dx%3D0)
The second line integral now becomes,
![l_2=\int\limits^5_0 {(4)y(0)+4^2dy} \,=0](https://tex.z-dn.net/?f=l_2%3D%5Cint%5Climits%5E5_0%20%7B%284%29y%280%29%2B4%5E2dy%7D%20%5C%2C%3D0)
![l_2=\int\limits^5_0 {16dy} \,=80](https://tex.z-dn.net/?f=l_2%3D%5Cint%5Climits%5E5_0%20%7B16dy%7D%20%5C%2C%3D80)
We now evaluate the third line integral from
![(4,5)](https://tex.z-dn.net/?f=%284%2C5%29)
to
![(0,5)](https://tex.z-dn.net/?f=%280%2C5%29)
.
An equation of the straight line joining
![(4,5)](https://tex.z-dn.net/?f=%284%2C5%29)
and
![(0,5)](https://tex.z-dn.net/?f=%280%2C5%29)
in the xy plane is
![y=5](https://tex.z-dn.net/?f=y%3D5)
.
![\Rightarrow dy=0](https://tex.z-dn.net/?f=%5CRightarrow%20dy%3D0)
The third line integral now becomes,
![l_3=\int\limits^0_4 {x(5)dx+x^2(0)} \,](https://tex.z-dn.net/?f=l_3%3D%5Cint%5Climits%5E0_4%20%7Bx%285%29dx%2Bx%5E2%280%29%7D%20%5C%2C)
![l_3=\int\limits^0_4 {5xdx} \,=-40](https://tex.z-dn.net/?f=l_3%3D%5Cint%5Climits%5E0_4%20%7B5xdx%7D%20%5C%2C%3D-40)
We now evaluate the fourth line integral from
![(0,5)](https://tex.z-dn.net/?f=%280%2C5%29)
to
![(0,0)](https://tex.z-dn.net/?f=%280%2C0%29)
.
An equation of the straight line joining
![(0,5)](https://tex.z-dn.net/?f=%280%2C5%29)
and
![(0,0)](https://tex.z-dn.net/?f=%280%2C0%29)
in the xy plane is
![x=0](https://tex.z-dn.net/?f=x%3D0)
.
![\Rightarrow dx=0](https://tex.z-dn.net/?f=%5CRightarrow%20dx%3D0)
The fourth line integral now becomes,
![l_4=\int\limits^0_5 {(0)y(0)+0^2dy} \,=0](https://tex.z-dn.net/?f=l_4%3D%5Cint%5Climits%5E0_5%20%7B%280%29y%280%29%2B0%5E2dy%7D%20%5C%2C%3D0)
We now add all the line integrals to get,
![l=0+80+-40+0=40](https://tex.z-dn.net/?f=l%3D0%2B80%2B-40%2B0%3D40)
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](https://tex.z-dn.net/?f=%5Cint%5Climits_C%20%7BP%28x%2Cy%29dx%2BQ%28x%2Cy%29dy%7D%20%5C%2C%3D%5Cint%5Climits%20%5C%2C%20%5Cint%5Climits_D%20%28%7B%5Cfrac%7B%5Cpartial%20Q%7D%7B%5Cpartial%20x%7D%7D-%7B%5Cfrac%7B%5Cpartial%20P%7D%7B%5Cpartial%20y%7D%7D%29%20%5C%2CdA)
.
This implies that,
![\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, \int\limits^4_0 ({2x-x }) \, dxdy](https://tex.z-dn.net/?f=%5Cint%5Climits_C%20%7Bxydx%2Bx%5E2dy%7D%20%5C%2C%3D%5Cint%5Climits%5E5_0%20%5C%2C%20%5Cint%5Climits%5E4_0%20%28%7B2x-x%20%7D%29%20%5C%2C%20dxdy)
This simplifies to
![\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, \int\limits^4_0 ({x }) \, dxdy](https://tex.z-dn.net/?f=%5Cint%5Climits_C%20%7Bxydx%2Bx%5E2dy%7D%20%5C%2C%3D%5Cint%5Climits%5E5_0%20%5C%2C%20%5Cint%5Climits%5E4_0%20%28%7Bx%20%7D%29%20%5C%2C%20dxdy)
We evaluate the inner integral to get,
![\int\limits_C {xydx+x^2dy} \,=\int\limits^5_0 \, ({8 }) \, dy](https://tex.z-dn.net/?f=%5Cint%5Climits_C%20%7Bxydx%2Bx%5E2dy%7D%20%5C%2C%3D%5Cint%5Climits%5E5_0%20%5C%2C%20%28%7B8%20%7D%29%20%5C%2C%20dy)
We now integrate again, to obtain,