Calculus III (MATH 241)

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.

∫C xe^(-2x)dx + (x^4 + 2x^2y^2)dy
C is the boundary of the region between the circles x^2 + y^2 = 9 and x^2 + y^2 = 25.

Use Green's theorem to evaluate the line integral along the given positively oriented curve.

xe^(−2x) dx + 

x^4 + 2x^(2)y^(2)dy

  C is the boundary of the region between the circles

x^2 + y^2 = 9

and

x^2 + y^2 = 25

Use Green's Theorem to evaluate F*dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y - cos(y), x sin(y)), C is the circle (x - 3)^2 + (y + 5)^2 = 25 oriented clockwise

Use Green's Theorem to evaluate F *dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y - cos(y), x sin(y)), C is the circle (x - 3)2 + (y + 5)2 = 25 oriented clockwise

Let W be the region within the cylinder x^2 + y^2 = 2 between the xy-plane and the cone z = √(x^2 + y^2). Set up the iterated integral of f(x, y) = x^2 + y^2 over W using... (a) ...rectangular coordinates. (b) ...cylindrical coordinates. (c) ...spherical coordinates. Evaluate one of the three integrals (your choice!).

Consider the solid E = {(x, y, z)| x^2+y^2+z^2 ≤ 2z} with the density function f(x, y, z) = √(x^2 + y^2 + z^2). (a) Use spherical coordinates to find the total volume of E. (b) Use spherical coordinates to find the total mass of E. (c) Find the average density over E.