Finding the volume surrounded by a curve using polar coordinate

[devinaxxx]

Homework Statement

I tried to answer the following questions is about the curve surface $z= f (x, y) = x^2 + y^2$ in the xyz space.

And the three questions related to each otherA.)

Find the tangent plane equation at the point $(a, b, a^2+ b^2)$ in curved surface z .

The equation of the tangent plane at the point $(a, b, f (a, b))$ on z is given by the following equation$Z-f(a,b)=f_x(a,b)(x-a)+f_y(a,b)(x-b)$So i got

$Z-(a^2+b^2)=2a(x-a)+2b(x-b)$

$2ax+2by-(a^2+b^2)$2)

when the tangent plane of the previous question moves pass through the point (0,0,-1). Find The equation for a plane S that contains the contact trajectory.Tried to put 0,0,-1 to equation in number 1

$-1=-(a^2+b^2)$

$z=1$ is S(?) But i wasnt so sure what is S plane here and what is the relation with Z?
3.

Calculate the volume V of the part surrounded by $z=x^2+y^2$ and the plane S
Note :

I was confuse about number 3, what is the area surrounded by S and Z (?)

Since i wasnt so sure about graph of S and Z here
can you help me to picture s and z? and also give me hint about the integration of the volume?

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

Homework Statement

I tried to answer the following questions is about the curve surface $z= f (x, y) = x^2 + y^2$ in the xyz space.
And the three questions related to each other
A.)
Find the tangent plane equation at the point $(a, b, a^2+ b^2)$ in curved surface z .
The equation of the tangent plane at the point $(a, b, f (a, b))$ on z is given by the following equation
$Z-f(a,b)=f_x(a,b)(x-a)+f_y(a,b)(x-b)$
So i got
$Z-(a^2+b^2)=2a(x-a)+2b(x-b)$

$2ax+2by-(a^2+b^2)$

You need $z =$ on the left side of that last equation although I think it is better to leave it in the form $$
z = a^2 + b^2 + 2a(x-a) + 2b(y-b)$$

2)
when the tangent plane of the previous question moves pass through the point (0,0,-1). Find The equation for a plane S that contains the contact trajectory.
Tried to put 0,0,-1 to equation in number 1
$-1=-(a^2+b^2)$
$z=1$ is S(?)
But i wasnt so sure what is S plane here and what is the relation with Z?

What you have shown is that the tangent plane will pass through $(0,0,-1)$ whenever $a^2+b^2=1$. In other words, if you take any point on the unit circle in the xy plane, the tangent plane to the paraboloid above that point will pass through that point. So there are lots of tangent planes that will work. You need more information to get a single plane S.

3.

Calculate the volume V of the part surrounded by $z=x^2+y^2$ and the plane S

You clearly need more information to get a single tangent plane, and even then you need a better description of what volume you are talking about. Also, in part 2, how does a tangent plane "move" and what do you mean by "contact trajectory"?

 

[MidgetDwarf]

Best way is to draw it out if you do not see it from the given equations. You have 3 cases: When Z=0, when X=0, when Y=0. Now graph.

Do not forget to take both x and y are equal to 0.

This gives you something to work with. Now expand this idea to your given problem.