(a) [10 pts] Let F = [x, xy], C: r = [2cost, 2sint], calculate the integral...

 

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(a) [10 pts] Let F = [x², xy], C: r = [2cost, 2sint], calculate the integral F(r) dr for 0≤t≤ π. (b) [10 pts] Let F = [x²y, -y³], C: r = [t, t"], (i) calculate these integrals F(r) dr based on (x, y) from (0, 0) to (1, 1) in terms of n; (ii) work out the values of the above line integrals by setting n = 1,2 and +00; (iii) based on result from (ii), state whether this line integral shows path independence property or not. ("yes" or "no") (a) [10 pts] Let F = [x², xy], C: r = [2cost, 2sint], calculate the integral F(r) dr for 0≤t≤ π. (b) [10 pts] Let F = [x²y, -y³], C: r = [t, t"], (i) calculate these integrals F(r) dr based on (x, y) from (0, 0) to (1, 1) in terms of n; (ii) work out the values of the above line integrals by setting n = 1,2 and +00; (iii) based on result from (ii), state whether this line integral shows path independence property or not. ("yes" or "no")