Evaluate the triple integral 0int1 0int20intx z dz dy dx

Question

1. Evaluate the triple integral ₀∫¹ ₀∫²₀∫^x z dz dy dx.

2. Evaluate∫∫∫_T z dV where T is the solid bounded by z = 0, and z = √(9-x²-y²).

3. Evaluate the triple integral ∫∫∫_T dV using spherical coordinates if T is the solid beneath the sphere x²+y²+z² = 4 and above the half cone z = √(x²+y²).

4. Evaluate the triple integral ₀∫¹₀∫^√(1-y2) _(x2+y2)∫ ^√(x2+y2) xyz dz dx dy.

5. Sketch the vector field given by F^- = (y - x) and explain si.fr) the sketch shows the field is not conservative.

6. Evaluate the line integral ∫_C 2x ds where C: x = t, y = t² with t ∈ [0, 1].

7. Compute the work done in moving an object through the field F^- = (2x + 3y, 3x - 3y²) from (0, 1) to (1, 0) along any path by finding a potential function and using the Fundamental Theorem.

8. Use Green's Theorem to evaluate ∫_C x⁴ dx + xy dy where C is the triangular curve consisting of the line segments from (0, 0) to (1, 0), from (1, 0) to (0, 1), and from (0, 1) to (0, 0).

 9. Evaluate the surface integral ∫∫_S z dS where S is the part of the plane z =1 + x bounded by the cylinder x² + y² = 1.

10. Compute the flux out of the sphere x² + y²+ z² = 1 if F^- = (y, x, z).

11. If F^- = (xz, yz, xy) and S is that pan of the sphere x² + y² + z²= 4 that lies inside the cylinder x² + y² =1 and above the x, y plane, use Stokes' Theorem to evaluate ∫∫_S Curl F^-•dS.