Question

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.

Answer 1

Step 1:

The integral is and vertices of the triangle are .

(a)

The integral is .

Graph :

(1) Draw the coordinate plane.

(2) Plot the vertices .

(3) Connect the plotted vertices to a smooth triangle.

Use .

Step 2:

Consider.

Observe the graph, the curve is bounded from .

Here coordinates are equal then the line is parallel to axis.

Since , then .

The limits of x are varying from 0 to 1.

.

Step 3:

Consider.

Observe the graph, the curve is bounded from .

Here coordinates are equal then the line is parallel to axis.

Since , then .

The limits of y are varying from 0 to 2.

.

Answer 2

Contd....

Step 4:

Consider.

Observe the graph, The curve is bonded from .

Using two points form of a line equation is .

Substitute in the line equation.

.

The limits of x is varying from 1 to 0.

Substitute in.

.

From .

.

Solution :

.

Answer 3

Step 1:

(b)

The integral is and vertices of the triangle are .

Greens theorem :

If C  be a positively oriented closed curve, and R be the region bounded by C, M and N are

the partial derivatives on an open region then.

Graph :

(1) Draw the coordinate plane.

(2) Plot the vertices .

(3) Connect the plotted vertices to a smooth triangle.

Step 2:

Observe the graph :

The limits of x are varying from 0 to 1 , so .

Find the bounds for y :

Lower limit :

Consider the points .

Here coordinates are equal then the equation of the line parallel to axis.

So the equation of the line is .

Lower limit of y is .

Upper limit :

Consider the points .

Using two points form of a line equation is .

Substitute in the line equation.

Upper limit of y is .

The limits of y is to , then .

Answer 4

Contd...

Step 3:

Using greens theorem,

The region bounded by the triangle is .

.

Solution :

.