The area of an ellipse is given by $Area=a\cdot b\cdot\pi$. (Compare to $\pi r^{2}$ for a circle.) Conceptually, integrating to get the area under the ellipse is not different from any other curve integration. We have to be sure that the curve doesn't double back on itself, but if we start with an ellipse in standard form and only use the part above the $x$-axis, then everything will work out. $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \tag{1} \label{1}$$ Solve $\eqref{1}$ for y to get the top half, $$y=\frac{b\sqrt{a^{2}-x^{2}}}{a}$$ and integrate from the left to right vertex. $$\intop_{-a}^{a}\frac{b\sqrt{a^{2}-x^{2}}}{a}dx=\frac{1}{2}ab\pi.$$ Thus the area of the complete ellipse is $\text{Area}=ab\pi$. Alternatively, we could start with the parametric equations: $$c=\left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} a\cos(t)\\ b\sin(t) \end{array}\right)$$ and using parametric integration under the curve, $$\intop_{-\pi}^{\pi}c\cdot dt=y(t)\cdot x^{\prime}(t)\cdot dt$$ Here $$y(t)=b\sin(t)\text{ and }x^{\prime}(t)=-a\sin(t)$$ so again, the top half, $$\intop_{-\pi}^{\pi}\left(b\sin(t)\right)\left(-a\sin(t)\right)dt=-ab\intop_{-\pi}^{+\pi}\sin^{2}t\,dt=\frac{1}{2}ab\pi.$$ Unfortunately, both of these integrals are pretty messy with the second one being slighty more familiar than the first. Either way, we will not show the detailed integrations here. Anyone interested in that detail can probably work it out for themselves.
The circumference is an issue. It can be approximated numerically by a few methods, but no exact algebraic form is known. For further information look up “elliptic integrals”. Later in this chapter we will look at finding the arc length of any curve for which we have an equation. That is done by integration too. For the ellipse, the integral is historically difficult (famous). It is called the complete integral of the second kind and as far as I know, third grade solutions are not available. However, that being said, an example problem to find the circumference is worked in the section on arc length.