Answer to Question #222758 in Calculus for Anuj

Question #222758

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


1
Expert's answer
2021-09-15T02:58:28-0400

b)

z=x2+y2

zx=2x,zy=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}^{2\u03c0}\\int_{1}^{\u221a3}\\sqrt(1+4r^{2})rdrd\\theta"

="\\int_{0}^{2\u03c0}d\\theta\\int_{1}^{\u221a3}\\sqrt(1+4r^{2})rdr"

Let 1+4r2=t2

8rdr=2tdt

"\\therefore" (2π-0)"\\int_{\u221a5}^{\u221a13}t*\\frac{tdt}{4}"

="\\frac{2\u03c0}{4}[\\frac{t^{3}}{3}]_{\u221a5}^{\u221a13}"

="\\frac{\u03c0}{6}(13\u221a13-5\u221a5)" square units


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS