Chapter 5 - Applications Of The Definite Integral In Geometry, Science, And Engineering - 5.1 Area Between Two Curves - Exercises Set 5.1 - Page 354: 40

$$\pi a b$$

Solve the equation to $y$. Remember that the area between the upper half and the $x$ -axis is twice the area of the ellips. \begin{array}{c} b \sqrt{1-\frac{x^{2}}{a^{2}}}=y \\ \text{Root} \\ -a=x \text { , } a=x \\ \text{Write integral of area} \\ 2 \int_{-a}^{a}\left(b \sqrt{1-\frac{x^{2}}{a^{2}}}\right) d x \\ \pi a b=2 \cdot \frac{\pi a b}{2} \end{array}