Question

Evaluate the line integral, where C is the given curve. C sin(x) dx + cos(y) dy, where C consists of the top half of the circle x2 + y2 = 16 from (4, 0) to (−4, 0) and the line segment from (−4, 0) to (−5, 4)

Answer 1

Parameterize the circular part of
`C` (call it
`C_1`) by

`x=4\cos t`

`y=4\sin t`

wih
`0\le t\le\pi`, and the linear part (call it
`C_2`) by

`x=-4-t`

`y=4t`

with
`0\le t\le1`.

Then

`\displaystyle\int_C\sin x\,\mathrm dx+\cos y\,\mathrm dy=\left\{\int_(C_1)+\int_(C_2)\right\}\sin x\,\mathrm dx+\cos y\,\mathrm dy`

`=\displaystyle\int_0^\pi(-4\sin t\sin(4\cos t)+4\cos t\cos(4\sin t))\,\mathrm dt+\int_0^1(-\sin(-4-t)+\cos4t)\,\mathrm dt`

`=0+\displaystyle\int_0^1(\sin(t+4)+\cos4t)\,\mathrm dt`

`=\cos4-\cos5+\frac{\sin4}4`