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Evaluate the surface integral $$\iint_S (x^2yz)\,\mathrm ds,$$ where $S$ is the part of the plane $z=y+3$ that lies inside the cylinder $x^2+y^2=1$. I'm very shaky on how to visualize the area we wish to integrate under and then bound the integral. Any help would be appreciated.

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  • $\begingroup$ Determine z limits then make a projection to the xy plane which will be the unit circle $\endgroup$ – Ameryr Mar 31 at 22:44
  • $\begingroup$ that's exactly what I'm asking about.. I was struggling to find the limit boundaries $\endgroup$ – joseph Apr 1 at 19:51
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Replace $z$ in the integral $$\int \int x^2 y(y+3) ds $$ $$\int \int x^2 y(y+3) \sqrt{1+f_x(x,y)+f_y(x,y)} ds$$ note $f(x,y)=y+3$, $y$ is changing from $-\sqrt{1-x^2}$ to $\sqrt{1-x^2}$ the $x$ limits are -1 to 1 https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/surfint/surfint.html

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