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 to question 1

The given line integral is

We evaluate the first line integral from to .

An equation of the straight line joining and in the xy plane is .



The first line integral now becomes,



We evaluate the second line integral from to .

An equation of the straight line joining and in the xy plane is .



The second line integral now becomes,





We now evaluate the third line integral from to .

An equation of the straight line joining and in the xy plane is .



The third line integral now becomes,





We now evaluate the fourth line integral from to .

An equation of the straight line joining and in the xy plane is .



The fourth line integral now becomes,



We now add all the line integrals to get,



Answer to question 2

According to the Green's Theorem,

.

This implies that,



This simplifies to  



We evaluate the inner integral to get,



We now integrate again, to obtain,