We consider the region between two curves
and
.
First, we find the intersection points. Since two curves can be written as
and
, plugging the first one into the second one, we get
![{\displaystyle 1-y^{2}=5-2y^{2}\iff y^{2}=4\iff y=\pm 2.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/3b3d3aa7f7d4459fafa20764fba8efa4ac4816fe)
Therefore, the intersection points are
![{\displaystyle (-3,2)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/feda11a75c8bbe13b2fe8a19c73d1c0c3055b763)
and
![{\displaystyle (-3,-2)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/2a52497a2b2926773ef821af6eec1f72d23d9334)
.
On the other hand, according to the question, the curve
lies on the right side of
, so that
when
.
To summarize, the area between the two curves has to be an integral of the form
![{\displaystyle \int _{-2}^{2}|(5-2y^{2})-(1-y^{2})|\,dy=\int _{-2}^{2}(5-2y^{2})-(1-y^{2})\,dy=\int _{-2}^{2}4-y^{2}\,dy=2\int _{0}^{2}4-y^{2}\,dy.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/0dfabd226987836a67042ad68626114c92486858)
The last equality follows from the fact that
is even. Therefore, the answer is
.
Answer:
.