Reference no: EM132599848

Question 1: Evaluate ∫_c ∇^→ f.dr. where (x, y, z) = 3x+2y/z+6 and is given by

r^→(t) = 3ti^+ 4j^+ (5 - t³)k^ with 1 ≤ t ≤ 2.

Question 2: Use Greens' Theorem to evaluate ∫_c (y² - 6y)dx + (y³ + 10y²) where C consist of the line segment from (-1,1) to (-1, -2) followed by the portion of the curve y = x² - 3 from (-1, -2) to (1, -2), which is in turn followed by the line segment from (1, -2) to (1,1) and finally by the portion of the curve = 2 - 2 from (1,1) to (-1,1).

Question 3: (a) Write down a set of parametric equations for the given surfaces
(i) The cylinder y² - 2y + z² = 3 for 2 ≤ x ≤ 5.
(ii) The portion of the sphere of radius 3 that is centred at origin with y ≥ 0 and z≤ 0.
(b) Find the equation of the tangent plane to the surface with parametric equations
x = u², y = v², z = u+ 2v at (1,1,3).

Question 4:  Find the surface area of the portion of the sphere of radius 4 that lies inside the cylinder x² + y² = 12 and above the xy-plane.

Question 5: Evaluate ∫∫_s (y +z ) where is the part of the cylinder x² + y² = 4 between the plane z = 0 and z = 4 - y.

Question 6: Evaluate ∫∫_s F^→ dS^→, where F^→(x,y,z) = -xi^+ 2yj^ -zk^ and is the portion of y = 3x² + 3z² that lies behind y = 6 oriented in the negative y-axis direction.

Question 7: Given F^→(x, y, z) = xi^+ yj^+2(1-z)k^ use divergence theorem to find the flux of F^→ across S, where S is the surface of the solid bounded by paraboloid x² + y² + z = 2 oriented upwards and the plane = 1 oriented downward.