Question

Let C be the boundary of the region bounded by x = y4 + 1 and x = 2, oriented in anticlockwise.

a) Calculate ∮(c) e^y/x dx + (e^y*lnx + 2x) dy using Green's Theorem.

Answer 1

Assuming the first term is
`\frac{e^y}x`, we have by Green's theorem,


`\displaystyle\int_C\left(\frac{e^y}x\,\mathrm dx+(e^y\ln x+2x)\,\mathrm dy\right)=\iint_R\left(\frac\partial{\partial x}\left(e^y\ln x+2x\right)-\frac\partial{\partial y}\left(\frac{e^y}x\right)\right)\,\mathrm dA`

`=\displaystyle\int_(y=-1)^(y=1)\int_(x=y^4+1)^(x=2)\left(\frac{e^y}x+2-\frac{e^y}x\right)\,\mathrm dx\,\mathrm dy`

`=\displaystyle2\int_(-1)^1\int_(y^4+1)^2\,\mathrm dx\,\mathrm dy`

`=\displaystyle2\int_(-1)^1(1-y^4)\,\mathrm dy`

`=\displaystyle4\int_0^1(1-y^4)\,\mathrm dy`

`=\frac{16}5`