Use green's theorem to compute the area inside the ellipse x252+y2172=1. use the fact that the area can be written as ∬ddxdy=12∫∂d−y dx+x dy . hint: x(t)=5cos(t). the area is 85pi .

b. find a parametrization of the curve x2/3+y2/3=42/3 and use it to compute the area of the interior. hint: x(t)=4cos3(t).

The area of the ellipse is given by

To use Green's theorem, which says

( denotes the boundary of ), we want to find and such that

and then we would simply compute the line integral. As the hint suggests, we can pick

The line integral is then

We parameterize the boundary by

with . Then the integral is

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Notice that kind of resembles the equation for a circle with radius 4, . We can change coordinates to what you might call "pseudo-polar":

which gives

as needed. Then with , we compute the area via Green's theorem using the same setup as before:

the answer is  

d.

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i did this one. so you should get it right.