# surface

1. ### How do I find the work done by the kinetic friction force when the surface is a curve

The problem is as follows: The diagram from below shows a boy of 30 kg of mass slides down a slide from a height of $5\,m$ starting from rest in point $A$. He reaches to point $B$ with a speed of $4\,\frac{m}{s}$. Find the work done by the frictional force in Joules. You may use...
2. ### Name of surface?

What is the name of a surface in hyperbolic space whose points are a fixed distance from a given plane in that space? Where can I find more information for such a surface?
3. ### Line, Surface, and Volume elements- Parameters

Line, Surface, and Volume elements. Let r be a vector in space r=(x,y,z) LINE x,y,z given in terms of parameter u. dr=r'du. dL=|r'|du SURFACE x,y,z given in terms of parameters u,v. In any coordinate direction, holding the other fixed, dr=r_{u}du, dr=r_{v}dv. dA=|r_{u}Xr_{v}|dudv...
4. ### Calculate the area of a quadrilateral

While thinking of strategies ... ;) I manage to solve the special case when M is on the half of the arc BC. The answer is: P_{APQD}=\frac{a^{2}}{2}. My reason was that because it does not matter where on arc BC is M, the area must be the same wherever it is positioned. I'm concerned that this...
5. ### Area of a Surface

I have solved this problem. I wish to find out if my solution is correct. **Problem:** Determine the area of the surface $A$ of that portion of the paraboloid: $$x^2+y^2-2z=0$$ where $x^2+y^2\le 8$, $y\ge x$ **Solution:** From the surface: $x^2+y^2-2z=0$...
6. ### Calculus 3 -Quadric Surface Question

Hi! I'm new here. I wanted help on a simple "Quadric Surface" Question from calculus 3. Our teacher in college didn't explained it in the class. Convert it into the standard form and sketch it: $x^2+y^2+z^2=3z$ Thanks!
7. ### Sine and Cosine to find area of surface

I am struggling with this one, finding the area of the land via the sine or cosine rule. I just don't see how I can go about it, I know I could use the formula for the area of a non-right angled triangle if I split it from B to D and make two triangles (Â½ ab sin C) However, I don't see how I...
8. ### surface integral problem

Find the area of the cylinder x^2 + z^2 = a^2 that lies inside the cylinder x^2 + y^2 = a^2. Official Answer: 8a^2 My solution x = r cos \theta , y = r sin \theta , z = \sqrt{a^2 - r^2cos^2 \theta } \int_{0}^{2 \pi} \int_{0}^{a} \sqrt { \frac {(r^2cos \theta sin^2 \theta + r^2 cos^3...
9. ### Proof of Surface Area and Volume of a sphere Using Integral Calculus

Hi! Letâ€™s consider a sphere with a radius r. What's his volume and his area. The full answer is given in the following link : https://smartmanmaths.com/2017/09/22/proof-of-surface-area-and-volume-of-a-sphere-using-integral-calculus/ Thanks and enjoy your mind!
10. ### Surface area via integration confusion

Hello forum, I am struggling on this problem. I am asked to evaluate $\oint \vec r \dot \,d\vec \sigma$ over the whole surface of the cylinder bounded by $x^2+y^2=1, z=0, z=3$ It seems pretty straight forward geometrically as it is just a unit circle at $z=0$ and then it extends...
11. ### 3-D Surface plot of equation

What does the plot of this surface equation look like ? I know it has a shape of the sphere but I am confused what is 1 and 4 and what they represent.
12. ### Curve vs Surface

I have a definition for curve and surface in my notebook that I don't really understand. Below are screenshots from the definitions and I would be really thankful if somebody explains them to me. Or you can explain to me what curve and surface is the way you know it. Thank you :D
13. ### parameterization in Surface integration

I want to surface integrate over a surface of the plane S : 2x+3y+6z =12 which lie in the 1st octant. Should I use parameterization, if I should, how?
14. ### requesting for explanation of surface integration

Evaluate âˆ«âˆ« < x, y, -2 > * N dS, where D is given by z = 1 - x^2 - y^2, x^2 + y^2 <= 1, oriented up. Official Answer: - pi Cross Product: \int_{0}^{2\pi} \int_{0}^{1} ( r cos \theta , r sin \theta , -2 ) * ( 2r^2 cos \theta , 2r^2 sin \theta , r ) ~ dr d \theta \int_{0}^{2\pi}...
15. ### Another Surface Integral problem... #2

Evaluate âˆ«âˆ« ( 2, -3, 4 ) * N dS, where D is given by z = x^2 + y^2, -1 <= x <= 1, -1 <= y <= 1, oriented up. Official Answer: 60 My solution: x = r cos \theta y = r sin \theta z = r^2 G(r, \theta) = ( r cos \theta , r sin \theta , r^2 ) dG/dr = ( cos \theta , sin \theta , 2r )...
16. ### Generic surface integral problem

Find the centroid of the surface of a right circular cone of height h and base radius r, not including the base. I don't even know how to find the two variables gradient of <x,y,z>..
17. ### Another Surface Integral problem...

Find the center of mass of an object that occupies the surface z = sqrt(x^2+y^2), 1 <= z <= 4 and has density z * x^2. Official Answer: ( 0, 0, 2275/682 ) Here is my solution: x = r cos \theta y = r sin \theta z = \sqrt { r^2 cos^2 \theta + r^2 sin^2 \theta } = r...
18. ### Flux of the vector field through the surface

Hello! I need help with determining the flux of the vector field. I'm enclosing link to my math.stackexchange question. https://math.stackexchange.com/questions/2382195/determine-the-flux-of-the-vector-field-trough-the-surface
19. ### Minimize surface area for tank

Hello forum. I am trying to make a rectangular tank that has a volume of 500 cubic feet, but minimizes surface area. The rectangular tank does not have a lid. $V = xyz = 500$ $\Rightarrow z = \frac{500}{xy}$ Surface area can be described as: $S = 2yz + 2xz + xy$ This accounts for the walls...
20. ### Tangent plane to surface

The questions says to find tangent plane to the surface x^2-y^2-3z=0 that goes through A(0,0-1) and is parallel to line : (0,0,0)+t(2,1,2). I'm really stuck here.