Parametric representations and surface area

LoffieZA

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May 13, 2020
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Hello, I am looking for someone to check my work for me. Please see the attached questions and my work. It deals with surface area and parametric representations.

Thank you
 

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Your solutions to problem 1 and 2.1 are correct. For 2.2 you have calculated "dS" using the length of the normal vector at the single point (4,2,0)\displaystyle (4, -2, 0) where u= 2 and v=π\displaystyle v= \pi. You can't do that. It has to be at any u, v. So dS=u2+4u2cos2(v)+4u2sin2(v)dudv=5u2dudv=u5dudv\displaystyle dS= \sqrt{ u^2+ 4u^2cos^2(v)+ 4u^2sin^2(v)}dudv=\sqrt{5u^2}dudv= u\sqrt{5}dudv. Yes, x(y2+z2)=2u(u2cos2(v)+u2sin2(v))=2u(u2)=2u3\displaystyle x(y^2+z^2)= 2u(u^2cos^2(v)+ u^2sin^2(v))= 2u(u^2)= 2u^3 (it seems more work that necessary to do it as xy2+xz2\displaystyle xy^2+ xz^2). So the integral is 2502π03u4dudv\displaystyle 2\sqrt{5}\int_0^{2\pi}\int_0^3 u^4 dudv.

For problem 3 you have one notational errors, perhaps a "typo". You have "dS=14\displaystyle dS= \sqrt{14}" when it should be, of course, dS=14dxdy\displaystyle dS= \sqrt{14}dxdy. But you do have the correct integral.

Problem 4 looks good.
 
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