Using Green’s Theorem to Evaluate Line Integrals

If we construct a closed curve C by piecing together a number of different curves end to end, the process of evaluating a line integral over C can be lengthy because there are so many different integrals to evaluate. If C bounds a region R to which Green’s Theorem applies, however, we can use Green’s Theorem to change the line integral around into C one double integral over R.

EXAMPLE 4: Evaluate the line integral

where C is the square cut from the first quadrant by the lines .

We can use either form of Green’s Theorem to change the line integral into a double integral over the square, where C is the square’s boundary and R is its interior.

1. With the Tangential Form Equation (3): Taking  and   gives the result:

2. With the Normal Form Equation (4): Taking , gives the same result:

EXAMPLE 5: Calculate the outward flux of the vector field 

across the square bounded by the lines  and .

Calculating the flux with a line integral would take four integrations, one for each side of the square. With Green’s Theorem, we can change the line integral to one double integral with , the square, and R the square’s interior, we have