Math 1

1. 4) Use Green’s theorem to evaluate the line integral
∮ ( 푥 + 푦2 ) 푑푥 + ( 1 + 푥 2 ) 푑푦
푐
Solution :
∮ 퐹1 푑푥 + 퐹2 푑푦
푐
= ∬(
푑퐹2
푑푥
−
푅
푑퐹1
푑푦
) 푑푥 푑푦
Limits :
0 ≤ x ≤ 1
푥 3 ≤ y ≤ √푥
=∬ ( 2푥 − 2푦 )푑푦푑푥 √푥
푥3
2
) 1
0
=∫ ( (2푥)(√푥) − √푥
− (( 2푥)(푥 3) − 푥 6)푑푥
3
2 − 푥 − 2푥 4 + 푥 6)푑푥 1
0
=∫ (2푥
= (4
5
푥
5
2 − 푥2
2
− 2푥5
5
+ 푥7
7
)
Substitute 0 and 1
= (4
5
(1
5
2 ) − 12
2
− 2(15)
5
+ 17
7
)
=
3
70