Question

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

Answer 1

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`

`=(15)/(16)(\sin5-\sin-4\cos5)\approx-2.7516`