# 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...