Surface integrals

Question

Evaluate

Answer 1

Step 1 :

If a smooth parametric surface S  is given by the equation

and S  is covered just once as ranges throughout the parameter domain D ,
then the surface area of S  is ,

where .

Step 2 :

Evaluate .

Here, ,

Step 3 :

.

Consider

.

Apply partial derivative on each side with respect to u .

.

Consider

.

Apply partial derivative on each side with respect to v .

.

Step 4 :

Thus, .

Solution :

.