Find the area of the surface. The helicoid (or spiral ramp) with vector equation r(u, v) = u cos(v) i + u sin(v) j + v k, 0 ≤ u ≤ 1, 0 ≤ v ≤ 7?.

Q: Evaluate the surface integral. y dS, S is the helicoid with vector equation r(u, v) = (u cos(v), u…

A: We have to evaluate the surface integral ∬SydS Given vector equation is r→u,v=ucosv,usinv,v…

Q: Use spherical coordinates. Evaluate ∭e ^ (x ^ 2 + y ^ 2 + z ^ 2) dV, where E is inside the sphere x…

A:

Q: The vector surface integral j L F • dS is equal to the scalar surface integral of the function…

A:

Q: Find the area of the surface of revolution r(u, v)= 6sin(u)cos(v)i+ 6uj+6sin(u)sin(v)k, where 0SusT…

A:

Q: Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u ER…

A:

Q: Verify the validity of the Stokes's theorem for the vector A = e-bz sin ø a, + pe-bz , over the open…

A:

Q: Find the surface area of the helicoid (spiral ramp) with vector equation T(u, v) = (u cos v, u sin…

A: We need to find the area of the surface:r→u,v=ucos v, u sin v, v ; 0≤u≤1, 0≤v≤3π

Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

A: The area of given helicoid is given by using double integral as follows

Q: Evaluate fff 1- x2 - y2-z2 dx dy dz where V is the volume of the sphere x? + y? + z2 = a? by…

A:

Q: Find the work Fdr done by the force F i+ yj+k around the curve C that is formed by the intersection…

A:

Q: Calculate the vector area of the helicoid parameterised as follows: F(t, s) = (t cos s, t sin s, s)…

A: Please see the below pictures for detailed solution.

Q: Find the area of the surface with the following parametrization: u COS V Τ (u, υ) - u sin v u? 0<u<…

A:

Q: Find a vector function that represents the cur ve of intersection of the paraboloid z a + mnd the…

A: Given that, the paraboloid is z=x2+y2 and the cylinder is x2+y2=22. The projection of the of the…

Q: | Calculate the surface integral /| G(r) dA, where G = (12 the parametrization r(u, v) = [3u, 2v,…

A: Introduction: We know that the formula for the surface integral is given by,…

Q: Evaluate the surface integral. y dS, S is the helicoid with vector equation r(u, v) = (u cos(v), u…

A: Consider the provided question,

Q: Describe the surface with the parametrization x(u, v) = cos(u), y(u, v) = sin(u), z(u, v) = for 0…

A: The parametrization, x(u, v) = cos (u), y(u, v) = sin (u), z(u, v) = v, Interval : 0≤u≤2π, v ∈ ℝ

Q: Given F(x, y, z) = x^2i + 2xyj + 3xzk and a region, being cut from the 1st octant by the sphere x^2…

A:

Q: Find an equation of the tangent plane to the given parametric surface at the specified point. r(u,v)…

A: Please see the explanation below

Q: | a2 dA, where D is the part of the unit disk where r+y 2 0.

A: To find :         ∬Dx2 dA where D is the part of the unit disk where x+y≥0

Q: Find an equation of the tangent plane to the surface represented by the vector-valued function at…

A:

Q: Find the center of mass, the moment of inertia about the coordinate axes, and the polar moment of…

A:

Q: Use cylindrical coordinates to evaluate SI vz dv where E is the solid E bounded by the cylinder,…

A:

Q: Evaluate Curlv•ñ , where v = 2xyi + (x² – 2 y)] + xzk and ñ is a unit vector normal to the surface…

A:

Q: Using Green’s theorem, evaluate the line integral ∮Cxydx+ (x+y)dy, where C is the curve bounding the…

A: Given: Where the components of the vector field are

Q: Calculate the surface integral // y ds, where S is the part of the helicoid defined by the S vector…

A:

Q: Evaluate the line integral ∫cF⋅dr, where F(x,y,z)=5xi+yj+3zk and C is given by the vector function…

A: we have given F(x,y,z)=5xi+yj+3zk r(t)=(sint,cost,t)=sint i+ cost j +t k⇒dr=(cost i -sint j…

Q: implement Stokes ' Theorem by taking the vector function Å = (x² + z)i + (y² + x)j+ (z² + y)k on the…

A:

Q: Verify Stokes' theorem for the vector ficld v=x'i+x' yj. C is the boundary of the rectangle whose…

A:

Q: Let F = . Use Stokes' Theorem to evaluate / curlF · d5, where S is the hemisphere x² + y² + z² =…

A:

Q: Evaluate F.N dS, where F(x, y, z) = 9x²i + 9xyj + 3x'y'k and S is the surface consisting of all…

A:

Q: Evaluate the line integral F. dr, where C is given by the vector function r(t). F(x, y) = xyi + 6y?j…

A:

Q: Find the surface area of the helicoid (spiral ramp) with vector equation r(u, v) = (u coS v, u sin…

A:

Q: Evaluate the surface integral [[ (a² + y²) dS where S is the surface with vector function r(u, v) =…

A:

Q: Given F(x, y, z) = xzi + 3yj + yzk be a vector function that passes through a surface S of z+x? + y?…

A:

Q: Evaluate the surface integral. y ds, S is the helicoid with vector equation r(u, v) = (u cos(v), u…

A:

Q: Use cylindrical coordinates to evaluate y dV where D is the solid that lies be- tween the cylinders…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u eR…

A:

Q: Use spherical coordinates to evaluate the triple integral / x² + y² + z² d mere E is the ball: æ² +…

A:

Q: The scalar component of a vector A normal to cylindrical surface of radius 2 at point p A = p'a, +…

A: Given that, A=pBAap+sinϕaϕ+2zaz And the cylinder  surface of radius 2 is, p=2p-2=0 Now the normal…

Q: Use spherical coordinates to evaluate the triple integral [[[ 2² + 3² + 2²dV, where E is the ball:…

A: We will first change it into spherical coordinate and then solve it.

Q: Given F(x, y, z) = -yzį + xzj – xyzk be a vector function that passes through a surface S of z +x? +…

A:

Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

A:

Q: Evaluate the surface integral.     Sy dS,  S is the helicoid with vector equation  r(u, v) =   u…

A: Given, r(u,v)=ucosv,usinv,v Now differentiate above with respect to u partially. We get,…

Q: Evaluate the surface integral. V1 + x2 + y2 dS S is the helicoid with vector equation r(u, v) = u…

A:

Q: Use Stokes theorem to evaluate: ∫∫Sc  n⋅(∇×F)dS, where F = 5y i − 5x j + 3⋅x^4⋅y^7 k, n is the…

A: Use Stokes theorem to evaluate:   ∫∫Sc n⋅(∇×F)dS,   where F = 5y i − 5x j + 3⋅x^4⋅y^7 k, n is the…

Q: ind a formula for a vector normal to the plane tangent to f(x,y) at (xo, Yo).

A:

Q: (b) Solve the electric flux through a paraboloid z = x² + y² – 16 that lies below plane z = 0. It is…

A:

Q: Q3 Evaluate Curlvoñ, where v = 2.xyi +(x² – 2 y)j + xzk and i is a unit vector normal to the surface…

A:

Q: Find the average x-coordinate of a point on the half-torus with the following parametrization (drawn…

A:

Q: For an area A in the x-y plane, the moment of inertia of the element dA about the x-axis is: O dlx =…

A: