# Evaluating the Surface Integral $\iint_S (x^2yz)$ d$s$

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.

• Determine z limits then make a projection to the xy plane which will be the unit circle – Ameryr Mar 31 at 22:44
• that's exactly what I'm asking about.. I was struggling to find the limit boundaries – joseph Apr 1 at 19:51

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