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4. Let (23+ y)i + (73 -2.1)j + (+3+3y)k, and let S be the outside surface of the portion of the solid sphere of radius 1 centered at the origin that lies in the first octant (see figure to the right). Give S the outward orientation. (a) Choose one appropriate theorem from the following list to calculate Ss dĀ: The Fundamental Theorem of Line Integrals, Green's Theo- rem, The Divergence Theorem, Stokes' Theorem. (Note: Please do not calculate the integral here; you'll do the calculation in part (d).) (b) If, instead of using the theorem you selected in part (a), we chose to calculate SsF.dĀ directly using brute force methods (from Sections 19.1 and 19.2), we would have to calculate several different integrals. How many different integrals would we need? Use a complete sentence to explain your answer. (Note: You are not being asked to set up or calculate any integrals in this part of the problem, just to say how many integrals would be required and to explain why.)

(c) Use the theorem that you chose in part (a) to calculate IsF.dĀ. Show your work, and circle your final answer.

125 19.1-19.2 Flur and Flux Integrals The Flux of a Constant Vector Field Through a Flat Surface F Fact. Let be a constant vector field, and let S be a flat surface. Then the flux off through S is given by Flux Of Through S where A is a vector that is normal to S in the direction of orienta tion and whose magnitude equals the area of S. We call à the area vector of S. Flux Integrals Goal: Devise a method for calculating the flux of a vector field F through a surface S that works even if F is not constant and/or S is not flat. FA -A

Definition. The flux integral of F through the oriented surface S is defined by F.dĀ lim Σ.ΔΑ. ||A||-+0 s