paraboloid

1. Find the shortest distance to the paraboloid

Find the shortest distance from the point $(0,0,b)$ to the paraboloid $z=x^2+y^2$. \begin{align*} D(x,y)&=\sqrt{x^2+y^2+(x^2+y^2-b)^2}\\ D_x&=\dfrac{2x+2(x^2+y^2-b)2x}{2D}\\ D_y&=\dfrac{2y+2(x^2+y^2-b)2y}{2D}\\ \end{align*} Setting the two derivatives equal to zero \begin{cases}...
2. Surface integral for paraboloid

Evaluate the surface integral $\iint_G (xz)dS$ Where G is the part of parabolid $z=x^{2}+y^{2}$ that lies in the first octant between planes z=4 and z=9 So I think I have this set up right as follows (after switching to polar coordinates): \$\iint (r^{4}cos\theta \sqrt{4r^{2}+1}) (dr...
3. use double integral to find volume of the solid bounded by the paraboloid & cylinder

use double integral to find volume of the solid bounded by the paraboloid z=x^2+y^2 above, xy plane below, laterally by circular cylinder x^2 +(y-1)^2 = 1 So, I broke it above and below y-axis, and used polar: r varies from 0 to 2sin(theta) and theta varies from 0 to pi. V = 2* integral from...
4. points of intersection between cone and paraboloid

Hey everyone, I need to find the points of intersection between the cone z=\sqrt(x^2+y^2) and z=2-x^2-y^2 So, what I did was 2-x^2-y^2=\sqrt(x^2+y^2) (2-x^2-y^2)(2-x^2-y^2)=x^2+y^2 x^4+y^4-5*x^2-5*y^2+2*x^2*y^2+4=0 The problem is, I don't know what to do for now, can any one give me any...
5. parameterization of a paraboloid.

Hi all, Could somebody please help me parameterize a paraboloid in cylindrical coordinates. Question is as follows: A paraboloid has height 2 and max radius 2. Based on cylindrical coordinates , enter the parameterization , such that and gives the paraboloid surface above. How do i do this...
6. prove paraboloid is umbilic by unit vectors

Prove that the point p=(0,0,0) on the paraboloid z= x² + y² is umbilic by computing k(û) for all unit vectors û ∈ T_p(M)
7. The equation of a hyperbolic paraboloid to derive the corner points of rectangle

Hi Folks, I have come across some text where f(x,y)=c_1+c_2x+c_3y+c_4xy is used to define the corner points f_1=f(0,0)=c_1 f_2=f(a,0)=c_1+c_2a f_3=f(a,b)=c_1+c_2a+c_3b+c_4ab f_4=f(0,b)=c_1+c_3b How are these equations determined? See attached pic Thanks
8. Find the volume bounded by the paraboloid.

Find the volume bounded by the paraboloid z=x^2 + y^2 +6 and the planes x=0, y=0, z=0, x+y=2. Would I be integrating this? I have no clue how to tackle this problem. I have looked through my textbook, searched through my personal notes, and even searched through the notes my professor uploads...
9. Point on paraboloid at which the tangent plane is parallel to plane

Find pt on paraboloid x = 5y^2 + 7z^2, if it exists, at which the tangent plane is parallel to plane -x + y + z = 3. Not completely sure how to approach this problem. I'm pretty sure it involves the gradient, so I set f(x,y,z) = x - 5y^2 - 7z^2 and found that gradient which was \nabla f = i -...
10. Intersection of paraboloid and plane

If C is the curve in reals^3 which is the intersection of the paraboloid {(x,y,z) s.t. z= (x^2) + (y^2)} With the plane {(x,y,z) s.t. z= 3 -2y} And L is the segment of C which lies in the half-space x>or=0. What is integral fds with bottom limit L (on integral) where f(x,y,z) := 5xy +xz?
11. perfect hyperbolic paraboloid cylinder

Find the equation for the surface of a Pringle (assuming it is a perfect hyperbolic paraboloid cylinder). Use cm for the units of distance. Include your measurements (to the nearest tenth of a cm), a diagram, and your computations.
12. surface area of a paraboloid

Hi, I tried to calculate the area of a paraboloid, say z=x^2+y^2, between z=0 and z=1. I thought that in order to do this calculation, I can think on the paraboloid as consisted of circles with radius sqrt(z), so the surface area would be given by the integral from 0 to 1 on 2*pi*sqrt(z) (the...
13. rate of change with a elliptic paraboloid

Can someone help me solve this? A point moves along the intersection of the elliptic paraboloid z=x^2+3y^2 and the plane x=2. At what rate is z changing with y when the point is at (2,1,7)?
14. Find point on paraboloid using gradient

"A normal line to the paraboloid z=8x^2+2y^2 also passes through the point (34,25,81). Find the point on the paraboloid that the normal line passes through." All I have been able to do is find the gradient. gradient = (16x)i+(4y)j Please help.
15. Question to do with volume of a solid between a paraboloid and a plane

Hi the question is Use polar coordinates to find the volume of the solid which is under the paraboloid z=x^{2}+y^{2}and above the disk x^{2}+y^{2}\leq9 in the plane z=0. thanks(Wink)
16. Volume under paraboloid and over a disk

Ok its a simple question really... say that I have to find the volume (using polar coordinates) of the solid under the paraboloid z=x^2+y^2 and above the disk x^2+y^2≤9. My approach would be to find the z value of where the cylinder and paraboloid intersect. Then find the volume of the...
17. Surface Area of a Circular Paraboloid

Find the surface area formula for a circular paraboloid with height h and radius r. It is formed by rotating y = (x^2)(h)/(r^2), 0 <= x <= r around the y-axis. So far I've come to here and am stuck:
18. how do you draw paraboloid

Triple Integral how do you draw paraboloid using z=1-x^2-y^2 sorry I haven't taken math for 2 years. I forget how to. I would really appreciated if you can explain me this is the whole problem use cylindrical coordinates. Evaluate ∫∫∫E (x^3 +xy^2)dV, where E is the solid in the first...
19. Plane Tangent to a Paraboloid.

- I need to find at what point on te paraboloid y=x^2+z^2 is the tangent plane parallel to the plane x+2y+3z=1. I can handle most of this problem myself, but I need to know if this is the correct equation to start with: f(x,y,z)=x^2-y+z^2 Then I would just apply...
20. Area of a portion of a paraboloid

Find the area of the portion of the paraboloid x=y^2+z^2 which is inside the cylinder y^2+z^2=9. My attempt : I notice that to be more friendly, I could change the problem as "the paraboloid x^2+y^2=z and the cylinder x^2+y^2=9", but I won't do that. So A=\iint _S \sqrt{ \left ( \frac{ \partial...