表面积

The area element

The graph formula, recovered

The sphere's area, by integration

A paraboloid's area, worked

A parametrization has $|\mathbf{X}_u \times \mathbf{X}_v| = 3$ at some point. A parameter rectangle of area $0.01$ sits above that point. What patch area does it cover, approximately?

The surface $z = 3x$ above the unit square has what area?

Where on the sphere's parameter rectangle does a unit of parameter area cover the least surface area?

A plane roof rises with slope $1$ in every direction (gradient magnitude $1$) over a floor of area $20$. Give the roof's area in the form a*sqrt(b).

The paraboloid $z = x^2 + y^2$ over the unit disk has area $\pi(5\sqrt{5} - 1)/c$. Give $c$, as a plain number.