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)

Accepted Solution

Parameterize the circular part of [tex]C[/tex] (call it [tex]C_1[/tex]) by[tex]x=4\cos t[/tex][tex]y=4\sin t[/tex]wih [tex]0\le t\le\pi[/tex], and the linear part (call it [tex]C_2[/tex]) by[tex]x=-4-t[/tex][tex]y=4t[/tex]with [tex]0\le t\le1[/tex].Then[tex]\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[/tex][tex]=\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[/tex][tex]=0+\displaystyle\int_0^1(\sin(t+4)+\cos4t)\,\mathrm dt[/tex][tex]=\cos4-\cos5+\dfrac{\sin4}4[/tex]