In order to find the area of an ellipse, we may make use of the idea of transformations and our knowledge of the area of a circle. For example, consider R\displaystyle {R}, the region bounded by the ellipse (x−2)249\displaystyle \frac{{\left({x}-{2}\right)}^{{2}}}{{49}} + (y+1)21\displaystyle \frac{{\left({y}+{1}\right)}^{{2}}}{{1}} = 1.

The easiest transformation to choose makes

u=\displaystyle {u}=   and v=\displaystyle {v}=  

which should be easily inverted to obtain

x=\displaystyle {x}=   and y=\displaystyle {y}=  

leading to a Jacobian of ∂(x,y)∂(u,v)=\displaystyle \frac{{\partial{\left({x},{y}\right)}}}{{\partial{\left({u},{v}\right)}}}=   .

And since ∫∫RdA=∫∫S∂(x,y)∂(u,v)dudv\displaystyle \int\int_{{R}}{d}{A}=\int\int_{{S}}\frac{{\partial{\left({x},{y}\right)}}}{{\partial{\left({u},{v}\right)}}}{d}{u}{d}{v} where the transformed region S is bounded by x2+y2=1\displaystyle {x}^{{2}}+{y}^{{2}}={1}, we calculate the area by multiplying the area and the Jacobian, arriving at  

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