Question

Find the length of l on the curve r(t)=5cos(3t)i-5sin(3t)j+3tk over the interval [3,7]

Answer 1

`\mathbf r(t)=5\cos3t\,\mathbf i-5\sin3t\,\mathbf j+3t\,\mathbf k`

`\mathrm d\mathbf r(t)=\mathbf r'(t)\,\mathrm dt=(-15\sin3t\,\mathbf i-15\cos3t\,\mathbf j+3\,\mathbf k)\,\mathrm dt`


`\displaystyle\int_C\mathrm d\mathbf r(t)=\int_(t=3)^(t=7)\|\mathbf r'(t)\,\mathrm dt\|=\int_(t=3)^(t=7)√(225\sin^23t+225\cos^23t+9)\,\mathrm dt`

`=\displaystyle3√(26)\int_(t=3)^(t=7)\mathrm dt`

`=12√(26)`