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 .
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.
.
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.
.
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 .
.
(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.
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 , so .
Using greens theorem,
The region bounded by the triangle is .
.
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