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1. 1 (8 points). Calculate the area of the quadrilateral with vertices at

(0,0), (4,2), (2,2), and (4,6)

An explicit line integral or sum of line integrals

equal to this area is:_________________________________

The area is: _______________________________

Solution. This can be done with Green's formula:

Area = Integral[(-y) dx} counterwise along the perimeter of the figure. The

given points are listed in counterclockwise order,

On the first leg, we can take x=4 t, y = 2 t, 0 <= t <= 1:
*In[1]:=*

** Int1 = Integrate[(-2 t)*4, {t,0,1}]**

*Out[1]=*

-4

On the second leg, we can take x=4 - 2 t, y = 2, 0 <= t <= 1:

*In[2]:=*

** Int2 = Integrate[(-2)*(-2), {t,0,1}]**

*Out[2]=*

4

On the third leg, we can take x=2 t + 2, y = 2+ 4 t, 0 <= t <= 1:

*In[3]:=*

** Int3 = Integrate[(-2 - 4 t)*(2), {t,0,1}]**

*Out[3]=*

-8

On the fourth leg, we return to the origin and can take x=4 - 4 t, y = 6 - 6 t,
0 <= t <= 1:

*In[4]:=*

** Int4 = Integrate[(-6 + 6 t)*(-4), {t,0,1}]**

*Out[4]=*

12

*In[5]:=*

** Area = Int1 + Int2 + Int3 + Int4**

*Out[5]=*

4

The explicit expression asked for is

Area = Integrate[(-2 t)*4, {t,0,1}]

+ Integrate[(-2)*(-2), {t,0,1}]

+ Integrate[(-2 - 4 t)*(2), {t,0,1}]

+Integrate[(-6 + 6 t)*(-4), {t,0,1}],

and its value is 4. You could equally well calculate the integral of x dy.

Up to Solutions to Test 4