a and b are the major and minor radii of the ellipse. You can find these easily from your parametric equations by finding the maximum values for x and y.
This is so because your ellipse is simply a stretching of a unit circle (x = cos t, y = sin t) along the axes. No rotation is involved.
Coordinate axes coincide with symmetry axes of the ellipse (see picture), ie сoordinate axes divide the ellipse into four equal parts. The fourth part of the required area , which is located in the first quadrant, we will find as the area of a curvilinear trapezoid, that adjacent to the X-axis:
Now we transform the integral to the variable , using ellipse parametric equations:
Find the lower integral limit:
If then
Find the upper integral limit [we take the value x = 9, because the maximum value of cosine is a unit]:
If then
So we have
Finally we have
See this picture