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

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

$\mathbf r(t)=\mathbf r'(t)\,\mathrm dt=t^5\,\mathbf i-t^4\,\mathbf j+t^3\,\mathbf k$
$\implies\mathrm d\mathbf r(t)=(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt$

$\displaystyle\int_C\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}f(x(t),y(t),z(t))\cdot\mathbf r'(t)\,\mathrm dt$
$=\displaystyle\int_0^1(3\sin(t^5)\,\mathbf i-\cos(t^4)\,\mathbf j-5t^8\,\mathbf k)\cdot(5t^4\,\mathbf i-4t^3\,\mathbf j+3t^2\,\mathbf k)\,\mathrm dt$
$=\displaystyle\int_0^1(15t^4\sin(t^5)+4t^3\cos(t^4)-15t^{10})\,\mathrm dt$
$=\displaystyle\int_0^1\bigg(3\sin(t^5)\,\mathrm d(t^5)+\cos(t^4)\,\mathrm d(t^4)-15t^{10}\,\mathrm dt\bigg)$
$=-3\cos(t^5)+\sin(t^4)-\dfrac{15}{11}t^{11}\bigg|_{t=0}^{t=1}$
$=-3\cos1+\sin1+\dfrac{18}{11}$