For this problem, let F^RightArrow = and S be the hemisphere z = SquareRoot 9 -

Question

For this problem, let F^RightArrow = and S be the hemisphere z = SquareRoot 9 - x^2 - y^2 with upward pointing normal. Stokes Theorem relates a line integral of a vector field to a surface integral of a different vector field. State precisely what Stoke rsquo s theorem says. Sketch S in space. find parametric equations of the curve C that is the boundary of surface S. Evaluate DoubleIntegral_s curlF^RightArrow MidDot dS^RightArrow by turning it into a line integral with Stoke rsquo s Theorem. (To be 100% clear. The integral you evaluate will be a line integral.) For this problem, let C be parameterized by x = 4 cos t, y = 4 sin t, z = 4 over t Belongsto [0, 2Pi] and F^RightArrow = Sketch C in space, showing the direction that it is traced. There are many possible surfaces that have C as a boundary... hemispheres, paraboloids, discs, etc. The simplest is probably a disc. Write out parametric equations for the disc with C as it rsquo s boundary. We rsquo ll call the disc S. Find the upward pointing unit normal of S Find the curl of F^RightArrow Evaluate conint F^RightArrow MidDot dr^RightArrow by turning it into a surface integral with Stoke rsquo s Theorem. (To be 100% clean The integral you evaluate will be a surface integral.)

Explanation / Answer

Integrate by parts. Z x sin x dx = x cos x + Z cos x dx = x cos x + sin x + c 2. Integrate by parts twice. Z x 2 e x dx = x 2 e x Z 2 x ex dx = x 2 e x 2x ex + Z 2 e x dx = x 2 e x 2x ex + 2 e x + c 3. Sustitute x = sin y. Z p 1 x 2 dx = Z cos y cos y dy = 1 2 Z (1 + cos 2y) dy = y 2 + 1 4 sin 2y + c = 1 2 x p 1 x 2 + 1 2 sin y +

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