Verify that the line integral and the surface integral of Stokes' Theorem are equal for the...

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Verify that the line integral and the surface integral of Stokes' Theorem are equal for the following vector field, surface S, and closed curve C. Assume that C has counterclockwise orientation and S has a consistent orientation. 2 F= (y - z,z-x, x- y); S is the cap of the sphere x +y +z = 16 above the plane z= V15 and C is the boundary of S. Construct the line integral of Stokes' Theorem using the parameterization r(t) = (cos t, sin t, y 15), for 0sts 2n for the curve C. Choose the correct answer below. A. |(-1+ V15 sint+ v 15 cos t) dt 2л В. | (1- V15 sint- V15 cos t) dt OC. |(- V15 sint- V15 cost) dt O D. (V15 sint+ 15 co cos t) dt Construct the surface integral of Stokes' Theorem using R= {(x,y): x +y <1} as the region of integration. Choose the correct answer below. 1 2n O A. 2rde dr 0 0 1 2n О В. - 4rd0dr 0 0 1 2n O c. . 4r de dr 0 0 1 2n O D. 2rde dr 0 0 Evaluate both integrals to verify that they are equal. What is the result? (Type an exact answer, using t as needed.) Verify that the line integral and the surface integral of Stokes' Theorem are equal for the following vector field, surface S, and closed curve C. Assume that C has counterclockwise orientation and S has a consistent orientation. 2 F= (y - z,z-x, x- y); S is the cap of the sphere x +y +z = 16 above the plane z= V15 and C is the boundary of S. Construct the line integral of Stokes' Theorem using the parameterization r(t) = (cos t, sin t, y 15), for 0sts 2n for the curve C. Choose the correct answer below. A. |(-1+ V15 sint+ v 15 cos t) dt 2л В. | (1- V15 sint- V15 cos t) dt OC. |(- V15 sint- V15 cost) dt O D. (V15 sint+ 15 co cos t) dt Construct the surface integral of Stokes' Theorem using R= {(x,y): x +y <1} as the region of integration. Choose the correct answer below. 1 2n O A. 2rde dr 0 0 1 2n О В. - 4rd0dr 0 0 1 2n O c. . 4r de dr 0 0 1 2n O D. 2rde dr 0 0 Evaluate both integrals to verify that they are equal. What is the result? (Type an exact answer, using t as needed.)