Calculus Answers

Fi‎nd the flu‎x of ->F (x, y, z) =〈4x, 3z + x^2, (y^2)/2> a‎cross the po‎sitively or‎iented su‎rface S giv‎en by ->R(u, v) = 〈2u, 4v, −u^2〉, 1 ≤ u^2 + v^2 ≤ 4,

F‎‎‎‎‎‎‎‎ind the mass of the lamina in the shape of the portion of the plane with equation 4x + 8y + z = 8 in the ‎‎‎‎‎first octant if the area density at any point (x, y, z) on the plane is δ(x, y, z) = 6x + 12y + z g/cm^2.

Find the work done in moving a particle along a curve from point A(1, 0, −1) to B(2, 2, −3) via the conservative force field ~F (x, y, z) = 〈2y³ − 6xz, 6xy² − 4y, 4 − 3x²〉.

(a) using the Fundamental Theorem for Line Integrals;

(b) by explicitly evaluating a line integral along the curve consisting of the line segment from A to P (1, 2, −1) followed by the line segment from P to B.

Find the area, take the elements of the area parallel to the x-axis. y= 2x³-3x³-9x; y=x²-2x²-3x.