Question

∫ 2y dx + 3x dy
where d is describe by 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1

Answer 1

Solution:

1) We need to verify that $\intop_c(2y)dx\;+\;(3x)dy=$ $\iint_D( \dfrac{∂}{∂x}(3x)-\dfrac{∂}{∂y}(2y) )dA$ , where D denotes the region bounded by C.

x = cost

y = sin t

$\int_C2ydx+3xdy=\int^{ \tfrac{\pi}{2} }_0[2sint(-sint)+3cost\;cost]dt=\dfrac{\pi}{4}$

$\iint_D( \dfrac{∂}{∂x}(3x)-\dfrac{∂}{∂y}(2y) )dA=\iint_DdA=\dfrac{\pi}{4}$