Question

Consider the solid region bounded above by z= * + and above by z = 20. Find the tangent plane, both normal vectors, and the curvature of the surface at (6,3, 1). Set up an integral and solve for the volume of this solid (Hint: use the transformation u = { and v = 3).

Answer 1

Consider the sole region bounted by and by 2=20 Z2 + 4 9 on vectors an the curvature of To find: Tangent plane a alarmas the serface at 16,3,1). z 1 = f(x, n 2) = z2 4 4 & fcant, tots <,!) <-- / - 1) - of(*=>]cu (5, 1), (-), 3:1) Eng the tangent plane: 7f (6,3,1) : <ub, -3, 2-1) 2o 2) 2-3,-1,1).<mo, 7-3, 21) 터 -3 (hr) - % (7-3) + 1(21) ze - 347 lg 3y + 2 + 2y = 2 2-34-227 Iz + 19 20 qrty z z 57 (eq. f the targent plane)

have, I of Cary, z) = (-7 %) & xf (6,3,1) (-3, -3,1) The normal line ir the line with F (94,34 (1,3,1) vel fino, y, z) - IF(8,3,1) that passes through the point (6,3,1) gn particular, The ea" (parametric) of the normal hne is . (t) Hot Fx (2o, 7o, to) = 1 r(+) = tot fy (x0, 2, 20)t Z(t) = E + Fz (no, n, 2a) + here, F (m, y, z) z z- ha fet 2-0 9 Fylgning mla -7 Dec,2.13 i Jerony Ez lei 1 n (t) = 6 -3t yet) = 3-%+ Z(t) = 1 +

curvature of the Serface Firy, 2) = Zn 4 To compute the mean & the Gaussian Cunraturex, we need If = < For Fiore >; <2, 1) O O H (f): for fry tuz fyn fyr fyz fer fer tzz O og odbora Trace (H(f)) = 3 2 4 4 4 2 Is 0 0 H* (f) 14 0 sex Hif) Gaussian curvatu DET kg If H* 1*+14 <a, D). 112

tas puery Mean curvature: Km If x H (f). If af f1² Trace (H(f)) 2 lef} kiz) - (ietery ( 2 Чт- 칠 4 + sy? 721 13 Чт + + 4 2 volume of this sela Fly, z) = z- 2.-* = 0 Taking 3. V egn becomes F (V, V, Z) - zute 1 bound b above hoy (Z - 20 So, z varies from o to 20 we have to evaluate the tripte integral Slzar SSS z tu dpt2 2

to 20 The upper ao lower liness of Setting z=0 we have ozenten را با & Z 24, ofenniana To determine limits of uur. The a limits of integration "-o to * * * V z trou & the r knins of integration -√2o-ur to troch The volume of the So lia is Vzoru 20 va mp up z p Z -ro w-u 2o-ur der av ara 20-4² de du -vou 200 no anorge -ro A

z4w Vrou au $ 4 z vo N ९ 2 I varat & su(2) ca] "Jeesus due *** q vwwt og med at [ of lo Sint (1) + Toocos lo font (13) [04 1068)* o -1010)] [i: Sall() = 1ų, 5*(4)943) [ 100%) + 10 (r)] to romano 4o 4w 2 40 lo > 4 wylot cubic units (4000 t