Set up and evaluate an integral to find the area enclosed by the ellipse
Find the roots of the ellipse. These will help you determine the bounds of integration.
Solve for y. This will be the integrand. Note: Because of symmetry across BOTH the x- and y-axes, you can find the area of half (or a fourth) of the ellipse, and then double (or quadruple) the result.
Use a calculator to evaluate the integral or graph the integrand and recognize its geometric shape.