Reference no: EM132599774

MATH251 Further Calculus and Introductory Analysis

Question 1: Find the surface area of the part of z = xy that lies in the cylinder given by x² + y² = 1.

Question 2: Determine the volume of the region that lies behind the plane x + y+ z = 2 and in front of the region in the yz-plane that is bounded by z = √y and z = y.

Question 3: Using the substitution u = 2x + 3y, v = x - 3y, find the value of the integral ∫∫_D e^2x+3y. cos(x - 3y) dA, where D is the region bounded by the parallelogram with vertices (0,0), (1, 1/3), (4/3 , 1/9), and (1/3 , - 2/9).

Question 4:

(a) Is there a vector field ^→G on R³ such that Curl ^→G = ?x sin y, cos y, z - xy? ? Explain.

(b) Verify that the vector field
^→F (x, y, z) = [2x cos(y) + 3]i^ + [2yez - x² sin(y)]j^ + y2ezk^

is conservative and determine its potential function.

Question 5: Given the vector field ^→F = ∇^→ ln(xy). Find the equipotential curves and the field lines for the vector field F . Sketch the equipotential curves.

Question 6: Evaluate ∫_c 6x ds where C is the portion of y= 3 + x² from x = -2 to x = 0 followed by the portion of y= 3 - x² from x = 0 to x = 2 which is in turn followed by the line segment from (2, -1) to (-1, -2).

Question 7: Find the work done by the force field ^→F = (yz + x)i^ + yzj^ + 3zk^ in moving the particle along the path given by r (t) = 3ti^ + 2 sin t j^ + 2 cos t k^, 0 ≤ t ≤ Π.