Question

Evaluate the line integral where f(x,y,z)=3sin(x)i−(cos(y))j−5xzk and c is given by the vector function r(t)=t5i−t4j+t3kr(t)=t5i−t4j+t3k , 0≤t≤10≤t≤1.

Answer 1

`\mathbf f(x,y,z)=3\sin x\,\mathbf i-\cos y\,\mathbf j-5xz\,\mathbf k`


`\mathbf r(t)=\mathbf r'(t)\,\mathrm dt=t^5\,\mathbf i-t^4\,\mathbf j+t^3\,\mathbf k`

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


`\displaystyle\int_C\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_(t=0)^(t=1)f(x(t),y(t),z(t))\cdot\mathbf r'(t)\,\mathrm dt`

`=\displaystyle\int_0^1(3\sin(t^5)\,\mathbf i-\cos(t^4)\,\mathbf j-5t^8\,\mathbf k)\cdot(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt`

`=\displaystyle\int_0^1(15t^4\sin(t^5)+4t^3\cos(t^4)-15t^(10))\,\mathrm dt`

`=\displaystyle\int_0^1\bigg(3\sin(t^5)\,\mathrm d(t^5)+\cos(t^4)\,\mathrm d(t^4)-15t^(10)\,\mathrm dt\bigg)`

`=-3\cos(t^5)+\sin(t^4)-(15)/(11)t^(11)\bigg|_(t=0)^(t=1)`

`=-3\cos1+\sin1+(18)/(11)`