Answer to Question #222758 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

b)

z=x²+y²

z_x=2x,z_y=2y

Change in polar coordinates

x=r Cos$\theta$

y=r Sin$\theta$

x²+y²=r²

dA=rdrd$\theta$

1<r<√3,0<$\theta$ <2$\theta$

$\sqrt (1+z_x^{2}+z_y^{2})=\sqrt(1+4x^{2}+4y^{2})$

=$\sqrt(1+4r^{2})$

Surface area=$\iint_R\sqrt(1+z_x^{2}+z_y^{2})dA$

=$\int_{0}^{2π}\int_{1}^{√3}\sqrt(1+4r^{2})rdrd\theta$

=$\int_{0}^{2π}d\theta\int_{1}^{√3}\sqrt(1+4r^{2})rdr$

Let 1+4r²=t²

8rdr=2tdt

$\therefore$ (2π-0)$\int_{√5}^{√13}t*\frac{tdt}{4}$

=$\frac{2π}{4}[\frac{t^{3}}{3}]_{√5}^{√13}$

=$\frac{π}{6}(13√13-5√5)$ square units