Question

Evaluate the line integral ∫cf⋅dr, where f(x,y,z)=3sinxi−4cosyj−xzk and c is given by the vector function r(t)=t5i−t4j+t3k , 0≤t≤1.

Answer 1

`\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j+z(t)\,\mathbf k`

`\mathrm d\mathbf r=(x'(t)\,\mathbf i+y'(t)\,\mathbf j+z'(t)\,\mathbf k)\,\mathrm dt`

`\implies\mathrm d\mathbf r=(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt`


`\mathbf f(\mathbf r(t))=3\sin t^5\,\mathbf i-4\cos(-t^4)\,\mathbf j-t^8\,\mathbf k`


`\mathbf f\cdot\mathrm d\mathbf r=15t^4\sin t^5+16t^3\cos t^4-3t^(10)`


`\displaystyle\int_(\mathcal C)\mathbf f\cdot\mathrm d\mathbf r=\int_(t=0)^(t=1)(15t^4\sin t^5+16t^3\cos t^4-3t^(10))\,\mathrm dt`

`=-3\cos t^5+4\sin t^4-\frac3{11}t^(11)\bigg|_(t=0)^(t=1)`

`=\left(-3\cos1+4\sin1-\frac3{11}\right)-\left(-3+0-0\right)`

`=4\sin1-3\cos1+(30)/(11)`