MA 227 Standard S10

Surface Integrals

At the end of the course, each student should be able to…

• S10: SurfInt. Compute and apply surface integrals.

S10: Surface Integrals

• The net volume of a solid with base given by the surface $$S$$ and heights given by $$f(x,y,z)$$ at each point is given by the surface integral $$\iint_S f\,d\sigma$$.
• When the surface is parametrized by $$\vect r(u,v)$$ with domain $$G$$ in the $$uv$$ plane, this integral may be computed as $$\iint_G f(\vect r(u,v))\|\vect r_u\times\vect r_v\|\,dA$$.
• The orientation of $$S$$ is irrelevant.
• The flux of the vector field $$\vect F$$ passing through the surface $$S$$ oriented by $$\vect n$$ is given by $$\iint_S \vect F\cdot\vect n\,d\sigma$$.
• When $$\vect r(u,v)$$ parametrizes $$S$$ with the correct orientation, then this integral may be computed as $$\iint_G \vect F(\vect r(u,v))\cdot(\vect r_u\times\vect r_v)\,dA$$.
• When $$\vect r(u,v)$$ parametrizes $$-S$$ (with opposite orientation), then this integral may be computed as $$-\iint_G \vect F(\vect r(u,v))\cdot(\vect r_u\times\vect r_v)\,dA$$.

Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 15.6 (exercises 1-36)