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

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

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

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

$=\cos4-\cos5+\dfrac{\sin4}4$