Do the following:
(1) Evaluate Int(P(x, y) dx + Q(x, y) dy) over the curve C, where P(x, y) = y^2, Q(x, y) = 3x, and C is the portion of the graph of the function y = 3x^2 from (-1, 3) to (2, 12). Here, "Int" stands for integral.
(2) Use the Divergence Theorem to evaluate the surface integral Int(F*n ds) over the surface S, where F = 3x i + 2y j + 4z k and S is the portion of the plane x + y + z = 6 that lies in the first octant. Here, "Int" stands for integral, F*n denotes the dot product of F and the unit vector n (the outward unit normal vector to S), and i, j, and k denote the unit vectors in the positive x, positive y, and positive z directions, respectively.
Complete, detailed solutions to both problems are provided in an attached .doc file (Calc1-WithSolutions.doc).
(1) The key step in solving problem 1 is to express every quantity in the integral in terms of the variable x (by using the fact that y = 3x^2 for every point on curve C), so that the ...
The given line integral and surface integral are evaluated, and a detailed explanation of each is provided.