Shawn Simard Assignment Section 13.6 due 12/01/2016 at 11:59pm MST 1. (1 point) Match the parametric equations with the verbal descriptions of the surfaces by putting the letter of the verbal description to the left of the letter of the parametric equation. 1. r (u, v) = ui + u cos vj + u sin vk 2. r (u, v) = ui + vj + (2u 3v) k 3. r (u, v) = u cos vi + u sin vj + u2 k 4. r (u, v) = ui + cos vj + sin vk A. circular paraboloid B. cone C. plane D. circular cylinder 7. (1 point) Find the surface area of the part p of the sphere x2 + y2 + z2 = 64 that lies above the cone z = x2 + y2 8. (1 point) Find the area cut out of the cylinder x2 + z2 = 4 by the cylinder x2 + y2 = 4. 9. (1 point) If a parametric surface given by r1 (u, v) = f (u, v)i + g(u, v)j + h(u, v)k and 4 u 4, 5 v 5, has surface area equal to 7, what is the surface area of the parametric surface given by r2 (u, v) = 3r1 (u, v) with 4 u 4, 5 v 5? 2. (1 point) Consider x = h(y, z) as a parametrized surface in the natural way. Write the equation of the tangent plane to the surface at the point (1, 4, 2) given that h y (4, 2) = 1 and h z (4, 2) = 3. Kim MAT 267 ONLINE B Fall 2016 10. (1 point) Parameterize the plane through the point (5, 5, 5) with the normal vector h5, 3, 1i ~r(s,t) = (Use s and t for the parameters in your parameterization, and enter your vector as a single vector, with angle brackets: e.g., as 1 + s + t, s - t, 3 - t .) . 3. (1 point) Consider the surface with parametric equations r(s,t) = hst, s + t, s ti. A) Find the equation of the tangent plane at (2, 3, 1). . 11. (1 point) For a sphere parameterized using the spherical coordinates and , describe in words the part of the sphere given by the restrictions B) Find the surface area under the restriction s2 + t 2 1 /2 0 4. (1 point) Use Equation 9 from section 13.6 to find the surface area of that part of the plane 5x + 6y + z = 3 that lies x2 y2 inside the elliptic cylinder + =1 81 64 and 2/3 5/4 0 /2. Then pick the figures below that match the surfaces you described. /2 0 : [?/1/2/3/4/5/6/7/8] 2/3 5/4 0 /2 : [?/1/2/3/4/5/6/7/8] Surface Area = 5. (1 point) Write down the iterated integral which expresses the surface area of z = y7 cos6 x over the triangle with vertices (1, 1), (1, 1), (0, 2): (Click on any graph to see a larger version.) Z b Z g(y) p a h(x, y) dxdy f (y) a= b= f (y) = g(y) = h(x, y) = 1. 2. 3. 4. 6. (1 point) The vector equation r (u, v) = u cos vi + u sin vj + vk, 0 v 5, 0 u 1, describes a helicoid (spiral ramp). What is the surface area? 5. 6. 7. 8. 1 12. (1 point) Consider the cone shown below. If the height of the cone is 3 and the base radius is 9, write a parameterization of the cone in terms of r = s and = t. x(s,t) = , y(s,t) = , and z(s,t) = , with s and t . c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 2