Review Problems for Calculus III Make Up Test 1

    Problem 1. Let u = (x³ /3) - x y², v = x²y - (y³ /3). If f_uu + f_vv = 0, does it follow that f_xx + f_yy = 0 ? Prove your assertion.

    Problem 2. Same problem as Problem 1, but with u = x/(x²+y²), v = -y/(x²+y²).

    Problem 3. Let z = f(x,y), and x = r cos q , y = r sin q , so that z = w(r,q ). Find w_r and w_q in terms of: f_x and f_y and r and q , but not x and y. Then solve for f_x and f_y in terms of: f_r and f_q and r and q .

    Problem 4. Set up two double integrals for the volume of space trapped between the paraboloid z = (x-1)²+y² and the plane 2 x + z = 5, one in terms of dx dy, and another in terms of dr and dq . Evaluate one of these integrals.

    Problem 5. Set up a double integral for the space trapped between the cone z = Ö[x²+ 2y²] and the cylinder x²+z² = 4x, either totally in x and y or totally in r and q . Don't evaluate.

    Problem 6.Do the same for the space trapped between the cone z = Ö[2 x²+y²] and the paraboloid 4x = z²+y² (whose axis of symmetry is the positive x-axis).

Also memorize my sample solutions for the volume of the sperm-whale, of the space between the two parabolic walls, z = 4-y² and z = x², and of the space between the cylinders x²+y²=a² and x²+z²=a².