Answer to Question #246092 in Calculus for Anuj

Question

Consider the surface S = n (x, y, z) | z = p x 2 + y 2 and 1 ≤ z ≤ 3 o .(a) Sketch the surface S in R 3 . Also show its XY-projection on your sketch. (2) (b) Evaluate the area of S, using a surface integral

Answer 1

a)

$z=\sqrt{x^2+y^2}$

XY-projection:

The XY-projection the sketch is the shaded region

b)

$x=zcos\varphi$

$y=zsin\varphi$

$z=z$

$r(z,\varphi)=zcos\varphi i+zsin\varphi j+zk$

$r_z=cos\varphi i+sin\varphi j+k$

$r_{\varphi}=-zsin\varphi i+zcos\varphi j$

$r_z\times r_{\varphi}=\begin{vmatrix} i & j&k \\ cos\varphi & sin\varphi &1\\ -zsin\varphi &zcos\varphi &0 \end{vmatrix}=-zcos\varphi i -zsin\varphi j+zk$

$|r_z\times r_{\varphi}|=z\sqrt 2$

surface area:

$S=\iint |r_z\times r_{\varphi}|dA=\sqrt 2\int ^{2\pi}_0\int^3_1 zdzd\varphi =8/\sqrt 2\int ^{2\pi}_0 d\varphi=8\pi\sqrt 2$