Question

Let F(x, y, z) = (5ex sin(y))i + (5ex cos(y))j + 7z2k. Evaluate the integral C F · ds, where c(t) = 8 t , t3, exp( t ) , 0 ≤ t ≤ 1. (Note that exp(u) = eu.)

Answer 1

I'm going to assume this reads

`\vec F(x,y,z)=5e^x\sin y\,\vec\imath+5e^x\cos y\,\vec\jmath+7z^2\,\vec k`

and the path
`C` is parameterized by

`\vec c(t)=8t\,\vec\imath+t^3\,\vec\jmath+e^t\,\vec k`

with
`0\le t\le1`. Under this parameterization,

`\vec F(x,y,z)=\vec F(x(t),y(t),z(t))=5e^(8t)\sin(t^3)\,\vec\imath+5e^(8t)\cos(t^3)\,\vec\jmath+7e^(2t)\,\vec k`

and

`\mathrm d\vec c=(\mathrm d\vec c)/(\mathrm dt)\,\mathrm dt=(8\,\vec\imath+3t^2\,\vec\jmath+e^t\,\vec k)\,\mathrm dt`

Then in the integral,

`\displaystyle\int_C\vec F\cdot\mathrm d\vec s=\int_0^1\vec F(x(t),y(t),z(t))\cdot(\mathrm d\vec c)/(\mathrm dt)\,\mathrm dt`

`=\displaystyle\int_0^1(7e^(3t)+15e^(8t)t^2\cos(t^3)+40e^(8t)\sin(t^3))\,\mathrm dt\approx\boxed{12586.5}`

(It's unlikely that an exact answer can be found in terms of elementary functions)