Concept explainers
(a)
To find:
An equation of the tangent plane at the point
Solution:
Explanation:
1) Concept:
The equation of the tangent plane is
Where
2) Given:
The parametric surface
3) Calculations:
The parametric surface
To find the tangent vectors
The normal vector to the tangent plane is
This can be written as,
At the point
For
The position vector of a point
The equation of the tangent plane is
This is the equation of the tangent plane
Conclusion:
Thus, anequation of the tangent plane at the point
(b)
To draw:
The graph of surface
Solution:
Explanation:
1) Concept:
To draw the graph of surface
2) Calculations:
The parametric surface
The equation of the tangent plane is
To draw the graph of surface
Conclusion:
The graph of surface
(c)
To set up:
An integral for the surface area
Solution:
Explanation:
1) Concept:
If a smooth parametric surface
And
Where
2) Given:
The parametric surface
3) Calculations:
The parametric surface
The surface area
Conclusion:
An integral for the surface area
(d)
To find:
Solution:
Explanation:
1) Concept:
2) Given:
3) Calculations:
From the part (a),
The surface integral is,
To find
Substitute this value insurface integral
Use the command in Mathematica
Therefore,
Conclusion:
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