With the curve [tex]C[/tex] parameterized by
[tex]C:\mathbf r(t)=\underbrace{15\sin t}_{x(t)}\,\mathbf i+\underbrace{13\sin^2t}_{y(t)}\,\mathbf j+\underbrace{12\sin^3t}_{z(t)}\,\mathbf k[/tex]
with [tex]0\le t\le\dfrac\pi2[/tex], and given the vector field
[tex]\mathbf f(x,y,z)=x\,\mathbf i+y\,\mathbf j+z\,\mathbf k[/tex]
the work done by [tex]\mathbf f[/tex] on a particle moving on along [tex]C[/tex] is given by the line integral
[tex]\displaystyle\int_C\mathbf f\cdot\mathrm d\mathbf r=\int\limits_{t=0}^{t=\pi/2}\mathbf f(x(t),y(t),z(t))\cdot\frac{\mathrm d\mathbf r(t)}{\mathrm dt}\,\mathrm dt[/tex]
where
[tex]\mathrm d\mathbf r=(15\cos t\,\mathbf i+26\sin t\cos t\,\mathbf j+36\sin^2t\cos t\,\mathbf k)\,\mathrm dt[/tex]
The integral is then
[tex]\displaystyle\int_0^{\pi/2}(15\sin t\,\mathbf i+13\sin^2t\,\mathbf j+12\sin^3t\,\mathbf k)\cdot(15\cos t\,\mathbf i+13\sin2t\,\mathbf j+18\sin t\sin2t\,\mathbf k)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^{\pi/2}(432\sin^5t\cos t+338\sin^3t\cos t+225\sin t\cos t[/tex]
[tex]=269[/tex]
