Quincy is trying to estimate the height of a tree in his backyard. he measures the tree’s shadow as 12 ft. he stands near the tr
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The general equation of an ellipse centered at the origin with its semiaxes coinciding with the coordinate axes is given by
Substituting
and 
into the above equation gives the Pythagorean identity, so we can use polar coordinates quite nicely to our advantage.
If
is the ellipse with the equation above, the area is given by the double integral
Let
and 
, so that the Jacobian matrix is
and the magnitude of its determinant is
since in polar coordinates we use the convention that
, and 
because they are lengths.
Now, the area is given by
Substituting
If
Let
and the magnitude of its determinant is
since in polar coordinates we use the convention that
Now, the area is given by