Answered

∫c

x sin y ds, C is the line segment from (0, 1) to (3, 5)

Verified Answer :

LammettHash

Parameterize the line segment \(C\) by

\(\mathbf r(t)=\langle x(t),y(t)\rangle=(1-t)\langle0,1\rangle+t\langle3,5\rangle=\langle3t,1+4t\rangle\)

where \(t\in[0,1]\). Then

\(\mathrm ds=\|\mathbf r'(t)\|\,\mathrm dt\)
\(\mathrm ds=\|\langle3,4\rangle\|\,\mathrm dt\)
\(\mathrm ds=5\,\mathrm dt\)

So the integral is

\(\displaystyle\int_Cx\sin y\,\mathrm ds=5\int_0^1x(t)\sin y(t)\,\mathrm dt\)
\(=\displaystyle5\int_0^13t\sin(1+4t)\,\mathrm dt\)
\(=\dfrac{15}{16}(\sin5-\sin-4\cos5)\approx-2.7516\)