Parametric representations and surface area

[LoffieZA]

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

 

[HallsofIvy]

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 $\displaystyle (4, -2, 0)$ where u= 2 and $\displaystyle v= \pi$. You can't do that. It has to be at any u, v. So $\displaystyle dS= \sqrt{ u^2+ 4u^2cos^2(v)+ 4u^2sin^2(v)}dudv=\sqrt{5u^2}dudv= u\sqrt{5}dudv$. Yes, $\displaystyle x(y^2+z^2)= 2u(u^2cos^2(v)+ u^2sin^2(v))= 2u(u^2)= 2u^3$ (it seems more work that necessary to do it as $\displaystyle xy^2+ xz^2$). So the integral is $\displaystyle 2\sqrt{5}\int_0^{2\pi}\int_0^3 u^4 dudv$.

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

Problem 4 looks good.

 

[LoffieZA]

Thank you!