Problems from Chapter 15.2: Math 293

Problems from Chapter 15.2 for Math 293

Return to the Main Math 293 Chapter 15 Section 2 Topics page Revised July 20, 2014
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Chapter 15
Section 15.2: Line Integrals

Problem 1

Evaluate the line integral `int_C 2xyz\quadds` for the line given by `r(t) = 3tbbi + 14tbbj + 18tbbk ` for `0<=t<=1`
We know that `int_C f(x,y,z)ds = int_a^b f(x(t),y(t),z(t)) sqrt( [x^'(t)]^2 + [y^'(t)]^2 + [z^'(t)]^2 ) \quad dt`.

thus for our problem
`int_C 2xyz\quadds = int_0^1 2(3t)(14t)(18t) sqrt( 3^2+14^2+18^2) \quaddt`
`int_C 2xyz\quadds = int_0^1 1512t^3 sqrt( 9+196+324) \quaddt`
`int_C 2xyz\quadds = 1512int_0^1 t^3 sqrt( 529) \quaddt`
`int_C 2xyz\quadds = 1512int_0^1 t^3 23 \quaddt`
`int_C 2xyz\quadds = 1512**23int_0^1 t^3 \quaddt`
`int_C 2xyz\quadds = 34776int_0^1 t^3 \quaddt`
`int_C 2xyz\quadds = 34776 ( t^4/4|_0^1 )`
`int_C 2xyz\quadds = 34776 (1/4-0/4 )`
`int_C 2xyz\quadds = 34776 (1/4 )`
`int_C 2xyz\quadds = 8694`

Problem 2

Find the mass of a wire with density `rho` where the equation of the wire is given as `r(t) = t^2bbi +2tbbj`, for `0<=t<=1` and `rho(x,y) = (3/4)y`
Then we have `r^'(t) = 2tbbi +2bbj` and `||r^'(t)||=sqrt(4t^2+4)`
and we know, in general,
`int_C bbF*dbbr = int_C bbF*bbT\quadds = int_a^b bbF(x(t),y(t))*bbr^'(t)\quaddt`
so
`int_C bbF*dbbr = int_Crho(x,y)\quad ds `
`int_C bbF*dbbr = int_C (3/4)y \quadds `
`int_C bbF*dbbr = int_0^1 (3/4)(2t)sqrt(4t^2+4)\quad dt `
`int_C bbF*dbbr = int_0^1 (3/4)(2t)2sqrt(t^2+1)\quad dt `
`int_C bbF*dbbr = int_0^1 3tsqrt(t^2+1)\quad dt `
`int_C bbF*dbbr = (t^2+1)^(3/2)|_0^1 `
`int_C bbF*dbbr = 2sqrt(2)-1 `

Problem 3

Find the mass of a wire with density `rho` where the equation of the wire is given as `r(t) = 2cos(t)bbi +2sin(t)tbbj + 3tbbk`, for `0<=t<=2pi` and `rho(x,y) = k + z` and `k>0`
Then we have `r^'(t) = -2sin(t)bbi +2cos(t)bbj+ 3bbk` and
`||r^'(t)||=sqrt((-2sin t)^2+(2cos t)^2 + 3^2)`
`||r^'(t)||=sqrt(4sin^2 t+4cos^2 t + 9)`
`||r^'(t)||=sqrt(4(sin^2 t + cos^2t+9)`
`||r^'(t)||=sqrt(4+9)`
`||r^'(t)||=sqrt(13)`
and we know, in general,
`int_C bbF*dbbr = int_C bbF*bbT\quadds = int_a^b bbF(x(t),y(t),z(t))*bbr^'(t)\quaddt`
so
mass = `int_C bbF*dbbr = int_Crho(x,y)\quad ds `
`int_C bbF*dbbr = int_C(k + z)\quad ds `
`int_C bbF*dbbr = int_0^(2pi)(k+3t)sqrt(13)\quad dt `
`int_C bbF*dbbr = sqrt(13)int_0^(2pi)k+3t\quad dt `
`int_C bbF*dbbr = sqrt(13)( kt +(3t^2)/2|_0^(2pi) ) `
`int_C bbF*dbbr = wsqrt(13)(2pik+6pi^2)`

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