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.

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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.

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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.

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The limits of x is varying from 1 to 0.

Substitute in .

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From .

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Step 5:

(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

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Graph :

(1) Draw the coordinate plane.

(2) Plot the vertices .

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

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 .

Therefore the limits of y is to , so .

Step 6:

Using greens theorem,

The region bounded by the triangle is .

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Solution :

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