Lateral Surface Area In Exercises 65–-72, find the area ofthe lateral surface (see figure) over the curve C in the x y-plane and under the surface
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- Dteermine the equation of the tangent plane to the surface given equation G(u, v) = (2u + v, u - 4v, 3u) at the point where: u = 1 and v = 4. %3Darrow_forwardR(a, b) = (2b + cosa, 2a + sin b, ab) Determine the equations for the (a) tangent plane and (b) the normal line to the surface S at the point (-1,2TT, 0)arrow_forwardSinx dA where R is the trangle in xy-plane bounded by the x-anise, the line y=x and. the line =arrow_forward
- Find the area of the surfaces The portion of the plane y + z = 4 that lies above the region cutfrom the first quadrant of the xz-plane by the parabola x = 4 - z2arrow_forwardTangent of x?/3 + y2/3 + z2/3 = a²/3 surface at any point ( xo , Yo ,Zo ) Show that the sum of the squares of the intersecting axes of the plane is constant.arrow_forwardQ1,- A- Locate the centroid (X only) of the shaded area y=12 -3x 3 m Fig. L.A 2 marrow_forward
- True False The equation of tangent plane to the surface f(x,y, z) = x² +y +z-9= 0 is x+y+z= 14. at the point (1,2,4) True Falsearrow_forwardderivative Q3: For the surface f(x, y, z)= In (xyz), find: (1) the gradient of f at (1,1,2), (2) the directional at (1,1,2) in the direction of i=i+2j-2k; (3) the max. and min directional derivative and their (4) the parametric equation of the tangent plane and the normal line to the surface at (1,1,2). directions;arrow_forwardCalculate the line integral of the vector field F = (y, x,x² + y² ) around the boundary curve, the curl of the vector field, and the surface integral of the curl of the vector field. The surface S is the upper hemisphere x² + y + z? = 25, z 2 0 oriented with an upward-pointing normal. (Use symbolic notation and fractions where needed.) F. dr = curl(F) =arrow_forward
- 38. Motion along a circle Show that the vector-valued function r(t) = (2i + 2j + k) %3D + cos t V2 j) + sin t V2 j + V3 V3 V3 describes the motion of a particle moving in the circle of radius 1 centered at the point (2, 2, 1) and lying in the plane x + y – 2z = 2.arrow_forwardParameterize the line of intersection of the surfaces: x2 - y2 = z - 1 and x2 + y2 = 4arrow_forwardCheck when the analytic function f(z) = sin z be a conformal mapping? %3Darrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage