2. A minimal surface is by definition a critical point of the area functional. Show that the graph of the function z(x, y)tany is a minimal surface. (This surface is called the helicoid.) 2. A m...
2. (20 marks) (a) Calculate the surface area of the graph of f(x,y) = x + 20y over the region R= {(x,y) e R2:1 < x < 4,2 sy s 2x} in the xy-plane. OV (b) Integrate the function g(x, y, z) = x +y +z over the surface that is described as follows: x = 2u – v, y = v + 2u, z= v – u Here u € [0,20), v € [0,21].
#P2 The differential of surface area, ds, for a surface determined by the graph of z=f(x,y) is calculated by dS = #P3 The differential of surface area, dS, for a surface determined by the graph of x = f(y,z) is calculated by dS = #P4 True or false: If Fis a velocity vector field for some fluid and S is a semipermeable surface, then the flux integral JJs FindS computes how quickly volume is passing through the surface S.
14. find surface area of S { (x,y,z) | x-3cost y-3sint z-z
14. find surface area of S { (x,y,z) | x-3cost y-3sint z-z
(2) The area of the surface with equation z = f(x,y). (x,y) E D. where fra f, are continuous, is A(S) = SVGC3. y)]? + [f;(x, y)]? +T dA If you attempt to use Formula 2 to find the area of the top half of the sphere x + y2 + 2? = a, you have a slight problem because the double integral is improper. In fact, the integrand has an infinite discontinuity at every point of the boundary circle...
Sketch the graph of the
functionf(x, y) = 3x+ 2y.Sketch the surface described by the
equationr−|z|= 0.Sketch the graph of the intersection of these two
surface
Sketch the graph of the function f(x, y) = 3x + 2y. Sketch the surface described by the equation r - |2-0. Sketch the graph of the intersection of these two surfaces.
Sketch the graph of the function f(x, y) = 3x + 2y. Sketch the surface described by the equation r - |2-0....
(1 point) Definition: The AREA A of the region that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles A = lim R, = lim [f(x)Ar + f(x2)Ax+... +f(x,y)Ax] 100 Wspacelin (a) Use the above definition to determine which of the following expressions represents the area under the graph of f(x) = x3 from x = 0 to x = 2. 64 A. lim 7100 11 i= B....
Find the values of x, y and z that correspond to the critical point of the function z = f(x,y) 3x2 + 5x + 5y + 2y?: = Enter your answer as a number (like 5, -3, 2.2) or as a calculation (like 5/3, 2^3, 5+4). T= Preview y= Preview z= Preview License Points possible: 10 Unlimited attempts.
2. Describe the graph of the following function: f(x,y, z)-2x + 3y + z 2.
2. Describe the graph of the following function: f(x,y, z)-2x + 3y + z 2.
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface area - f(x, y) ds z =f(x, y) Lateral surface xy) As C: Curve in xy-plane
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface...
Problem 1 Let gi(x, y, z)-y, 92(x, y, z)z and f(x, y, z) is a differential function We introduce F(x, y, z, A, )-f(x, y, z) - Xgi(x, y, z) - Hg2(x, y, 2). ·Show that the Lagrange system for the critical points off with constraints gi (x, y, z) = 92(x,y, z)0: F(zo, yo, 20, λο, μο)-(0, 0, 0, 0, 0) is equivalent to the one-dimensional critical point equation: df dr(ro, 0, 0) = 0, 30 = 20 =...