Finding the surface area defined by 2 expressions.

Question

How would I be able to determine the surface area definied by $z = xy$ and $x^2+y^2 \leq 1$.

Answer 1

The area of the surface will be given as the integral :

$$A=\int\int_S z(x,y)dS$$

Your area is defined by : $z=xy$ and the inequality $x^2+y^2 \leq 1$

This means that :

$$-1 \leq x \leq 1$$

$$-\sqrt{1-x^2} \leq y \leq \sqrt{1-x^2}$$

Thus, your surface area is :

$$A_z = \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-1}^{1}xydxdy$$

Note : You can reverse the integration order so that it will give you a numerical value.