SKKU-Calculus-Sec-15-6 Surface Area by SGLee with 김유경

15.6    Surface Area      by SGLee with 김유경

1-3.  Find a parametric representation of the surface.

1.

Let the parameters  and ,

we find that parametric surface is given by equations  and .

Therefore, the plane is described by the vector function

.

2. .

Let the parameters  and ,

we find that parametric surface is given by equations  and .

Therefore, the plane is described by the vector function

.

3. The portion  from  to .

4-5. Find an equation of the tangent plane at the point on the surface corresponding to the given parameter values.

4. .

At  and , the point on the surface is . The surface is defined by the vector function

, then

,   and

.

The normal vector  to the surface at  and . An equation of the tangent plane at that point is

or .

Note : 처음에 a,b,c를 x,y,z를 놓지않아야 한다. x,y,z라고 놓게 되면, tangent plane이 u,v에 관한 함수로 나오게 되어서 제대로 답을 구할 수 없다.

5. .

6-8. Calculate the area of the surface.

6. The surface with parametric equations .

and then form the cross product

The area is

.

7. The part of the plane  that lines in the first octant.

8. The part of the plane  tha lines inside the cylinder .