Question

Use Green’s Theorem to evaluate ∫c F⃗ ⋅ dr⃗

for the vector field
F⃗ =< e^x + x^2y, e^y − xy^2 > and C is the circle x^2 + y^2 = 25
orientated clockwise

Answer 1

`\displaystyle (625\pi)/(2)`

Explanation:

Recall the formula to apply Green's Theorem to a line integral:

`\displaystyle \oint_CPdx+Qdy=\iint_R\biggr(\biggr((\partial Q)/(\partial x)\biggr)-\biggr((\partial P)/(\partial y)\biggr)\biggr)dA`

Let the vector field
`F(x,y)=\langle P,Q\rangle` and calculate the curl of F:

`\displaystyle \\abla\cdot F = \biggr((\partial Q)/(\partial x) -(\partial P)/(\partial y) \biggr)\\\\\\abla\cdot F = -y^2-x^2`

Therefore, we have:

`\displaystyle -\iint_R(-y^2-x^2)dA\\\\=\iint_R(x^2+y^2)dA\\\\=\int^(2\pi)_0\int^5_0r^2r\,drd\theta\\\\=\int^(2\pi)_0\biggr((625)/(4)\biggr)d\theta\\\\=(625\pi)/(2)`