2. (20 marks) (a) Calculate the surface area of the graph of f(x,y) = x +...
8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29 8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29
Evaluate the surface integral lis(r,y,z) (x, y, z) ds where f(x, y, z) = x + y + z and o is the is the surface of the cube defined by the inequalities 0 < x < 5,0 Sy < 5 and 0 <3 < 5. [Hint: integrate over each face separately.] 1 f(x, y, z) ds =
Find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the integrals are simpler in polar coordinates.) f(x, y) = x2 + y2, R = {(x, y): 0 = f(x,y) 3}
(7 pts.) Let f(x, y, z) = "y and let R be the region {(x, y, z) |2 < x < 4,0 Sy < 3,15 zse}. 2 Evaluate | $180,0,.2) av. R
#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.
Using the change of variables u = x2y and v = y/x, integrate f(x,y) = x2y2 over the region bordered by y = 1/x2, y = 3/x2, y = x and y = 2x. 3. Using the change of variables u = ry and v = y/x, integrate f(x,y) = r2y2 over the region bordered by y=1/x?, y = 3/r?, y = r and y=2r.
7. Find the surface area of the surface r(u, u) = u ui + (u + u)j + (u-u) k, u2 +02-1 V/16-x2-y2 with upward orientation and let 8. Let S be the hemisphere 2 F(x, y,z)-yitj+3z k. Calculate JJs F dS, the flux of F across S 7. Find the surface area of the surface r(u, u) = u ui + (u + u)j + (u-u) k, u2 +02-1 V/16-x2-y2 with upward orientation and let 8. Let S be...
(20 points total) (a) Approximate to three decimal digits of the surface area patch on the graph zx3 Sin(y-1)+ ey-1, for 1 sx s 1.1, 0.8 sy (b) Let S be the surface described by fx2+ y 1;x 2 0:0 s z s (z + 1) ds 2) s1. s 2). Evaluate the integral (20 points total) (a) Approximate to three decimal digits of the surface area patch on the graph zx3 Sin(y-1)+ ey-1, for 1 sx s 1.1, 0.8...
Question 8: 10 Marks The area of the surface described by zf(x,y) for (r,)e R is given by Find an approximation to the area of the surface on the hemisphere r2 + y2 + z2-9,2 the region in the plane described by R-((r, yjo-r 1,0 y I) using: 0 that lies above 8.1 Trapezoidal rule in both directions 8.2 Simpson rule in both directions 8.3 Three-term Gaussian quadrature formulas in both directions 131 131 Question 8: 10 Marks The area...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...