Question

Imagine a field whose center is at the origin. There is a density of insects flying around given by the equation: s={ 4-Vx2 + y2 for x2 + y2 < 4 for x2 + y2 > 4 You begin at position (-1,-1) and your goal is to arrive at (1,1). There are two paths you are allowed to take: 1: The diagonal line connecting your start and end points 2: The semicircle connecting the start and end points at constant distance from the origin Determine which path causes you to run into fewer insects by calculating the line integrals.

Answer 1

Given 8 - Гц- у м+9 - 0 : м+*4ч ini+y?7,4 line lime connect - f n²ty2=2 connecting (-11) to (1,1) - Y+1 = f (gay) RCA 1 1 2 3 4 (-1,-1) Prca x=25-1 ey= 25-1 dn=adt dy=adt s dx = ss entertaines - 5' {u- 1 2 3* (232) 3] Jutust - 4 215 [1 - 2 - 344 = 5 252[u- 12(23-1)] dt - $ 253 [4- 2f2 4 +127 = 252 [ Auto2) +- Set] = 282 [ uth-52] = 882 = 11-3137 1. C = n²ty²=2 nz J2 cost, y=82 simt dn=-Sz Sintat, dy= 52costat t: -3tly to Mli.

Thy -3iTu _3174 = 5 4-52 ) J (asz simnet+ (Sz Cosby- de = 5 (u-52) Sz dt = 32 (n=92) ( + 3) ¿ J2 (4-52) T - 3T/u = 11-4883 the diagonal line semicirete path insects as to run into fewer causes 11.3137 < 11.4883