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parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
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Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ... 2answers 31 views Does g(x,y,z) (the equation of the surface) need positive z or negative z when doing a surface integral? \quadIf a smooth surface S is defined by g(x,y,z)=0, then recall that a unit normal is$$\mathbf{n}=\dfrac{1}{\|\nabla g\|}\nabla g,\tag{9}$$where \nabla g=\dfrac{\partial g}{\partial ... 1answer 36 views Computing the surface integral of a parabloid Problem: Solution: I am having difficulty understanding how the author determined the limits of integration of R. The author used \theta=\pi/3\quad to\quad \theta=\pi/2 and r=1\quad to\quad ... 1answer 51 views Find an integral for the area of the surface generated by revolving the curve y=sin(x) between 0 \le x \le \pi, about the x-axis So here is my problem: Find an integral for the area of the surface generated by revolving the curve y=sin(x) between 0 \le x \le \pi, about the x-axis Just thinking about the problem I feel ... 1answer 45 views Paremetric surface revolved around y-axis if I'm finding the area of the surface generated by revolving the curve around the y-axis I use the equation 2\pi x\sqrt{(x')^2+(y')^2} and I'm given$$x=(2/3)t^{3/2}y=2\sqrt{2}$$and I got ... 0answers 115 views Surface Integral The glass dome of a futuristic greenhouse is shaped like the surface z = 8 - 2x^{2} - 2y^{2}. The greenhouse has a flat dirt floor at z = 0 Suppose that the temperature T, at points in and around ... 1answer 117 views Recognize the equation of a surface of revolution Yesterday, I asked a question about the critic points of the surface$$z = (x^2 + y^2)e^{-(x^2 + y^2)}$$and my question was if I had a easier way to classify the critic points of this surfaces ... 2answers 125 views Maximum surface area of cylinder (1-variable) In a given sphere of radius R, it is required to find the cylinder with maximum surface area that we can inscribe in this sphere. Using that the radius of the cylinder is r, with Pythagoras ... 2answers 544 views Projection of ellipsoid Find the projections of the ellipsoid$$ x^2 + y^2 + z^2 -xy -1 = 0$$on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in ... 1answer 76 views Why xyz = e^x can be seen as the level surface f(x,y,z) = xyz - e^x? That does not make sense to me. I recognize a level surface from the form f(x,y,z) = k. Where is the k there? It looks just like a 3 variables function to me. 0answers 56 views Did I solve this surface area of revolution problem correctly y=\frac{1}{3}x^3, when 0 \le x \le 2 My Professor hasn't posted the solutions to our practice exam just yet, but I'd like to know if I solved the following surface area of revolution problem correctly? y=\frac{1}{3}x^3, when 0 \le x ... 1answer 154 views Show that 2 surfaces are tangent in a given point Show that the surfaces  \Large\frac{x^2}{a^2} + \Large\frac{y^2}{b^2} = \Large\frac{z^2}{c^2} and  x^2 + y^2+ \left(z - \Large\frac{b^2 + c^2}{c} \right)^2 = \Large\frac{b^2}{c^2} \small(b^2 + ... 0answers 49 views Gauss and Stocks teory Given \phi\in C^1(R), and we define the curve and surface \gamma={(x,y):y=\phi(x),0\le x\le 1} S={(x,y,z):z=\phi(\sqrt{x^2+y^2}),x^2+y^2\le 1} a.I need to prove that A(S)=2\pi\int_\gamma ... 0answers 83 views Working with projection of areas? I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ... 2answers 116 views Find a point through which every surface tangent to z=xe^(y/x) passes Find a point through which every plane tangent to the surface$$ z=xe^{\frac{y}{x}} $$passes. It's not a homework. I know, that I need a normal vector and the point of tangency to find a tangent ... 1answer 39 views The connectivity of the intersection of hypersurface and ball u is a function defined on a connected open set \Omega of \mathbb R^n containing 0 such that u \in C^2(\Omega) and u(0)=0. Consider the hypersurface X=\{(x,u(x))~|~x\in\Omega\} and the ... 1answer 131 views Surface Integral directly Surface integral that has me stumped. Q: Calculate \int \int_{S} F \cdot dA Where F(x,y,z)= xi+yj+zk S is the boundary of the region x^{2}+y^{2} \leq z \leq (2-x^{2}-y^{2})^{1/2} oriented so ... 2answers 1k views Find surface area that lies above a triangle Determine the area of the part of the surface z=2 + 7x + 3y^2 that lies above the triangle with vertices (0,0), (0,8), and (14,8). I do not know what formula to use to attempt this problem! 1answer 158 views Compute the surface area of an oblate paraboloid Consider the surface S: z=4-4x^2-y^2, z\geq0. Compute its surface area. I've tried the following: Area(S)=\int\int_D \sqrt{(8x)^2+(2y)^2+1}dxdy with D being the interior of the ellipse ... 0answers 37 views Why does the surface area integral need the arc length differential but the volume doesn't? [duplicate] When calculating the surface area of a revolution you need to use the arc length differential$$\sqrt{1 + y'^2}$$but you don't need to use that when calculating the volume. Why is that? Thanks! 1answer 38 views Approximated distance between two points on a surface I'm reading this paper and I don't understand this line because I haven't the book (I can't look up Theorem 7.4.2) . How come that the distance between P_i and P_{i-1} is calculated as follows? 1answer 60 views Area of a 3D surface I need to compute the area of a 3D sphere centered on 0;0;0 and the book I'm following says: "If a curve y = f (x) from y = a to y = b is revolved around the x axis, the surface area of ... 2answers 140 views Surface of revolution using cylindrical polars Consider the surface of revolution of the curve$$y = x^2$$where 0 < x < 1. By writing a suitable integral, show that the area of this surface is 3.81 units. (You are advised to work in ... 2answers 243 views How to parameterize a hyperboloid in a solid of revolution The middle “hyperboloid” part of the solid of revolution is determined entirely by a single edge of the cube that does not touch one of the axis vertices - there are six such edges. Mark these ... 0answers 204 views Understanding surface area of a revolution/length of curve I don't quite understand why the formula to find the surface area of a revolution is what it is:$$A = 2\pi \int_a^b x\ \sqrt{1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2}\ \text{d} x.$$I ... 2answers 2k views How to find surface area of x=\sqrt{a^2-y^2} I still hard time to find surface area of function... I have The given curve is rotated about the y-axis. Find the area of the resulting surface.$$x= \sqrt{a^2-y^2},\quad ...
I'm kind of confused about the explanation of the surface area formula in my text book The text gave us $$\int_{a}^{b}2\pi f(x) \sqrt{1+[f'(x)]^2}dx$$ after that the formula is getting like ...