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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 16.5 : Fundamental Theorem for Line Integrals
- Evaluate \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) where \(f\left( {x,y} \right) = 5x - {y^2} + 10xy + 9\) and \(C\) is given by \(\displaystyle \vec r\left( t \right) = \left\langle {\frac{{2t}}{{{t^2} + 1}},1 - 8t} \right\rangle \) with \( - 2 \le t \le 0\).
- Evaluate \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) where \(\displaystyle f\left( {x,y,z} \right) = \frac{{3x - 8y}}{{z - 6}}\) and \(C\) is given by \(\vec r\left( t \right) = 6t\,\vec i + 4\vec j + \left( {9 - {t^3}} \right)\vec k\) with \( - 1 \le t \le 3\).
- Evaluate \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) where \(f\left( {x,y} \right) = 20y\cos \left( {x + 3} \right) - y{x^3}\) and \(C\) is right half of the ellipse given by \(\displaystyle {\left( {x + 3} \right)^2} + \frac{{{{\left( {y - 1} \right)}^2}}}{{16}} = 1\) with clockwise rotation.
- Compute \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F = 2x\,\vec i + 4y\,\vec j\) and \(C\) is the circle centered at the origin of radius 5 with the counter clockwise rotation. Is \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) independent of path? If it is not possible to determine if \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) is independent of path clearly explain why not.
- Compute \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) where \(\vec F = y\,\vec i + {x^2}\,\vec j\) and \(C\) is the circle centered at the origin of radius 5 with the counter clockwise rotation. Is \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) independent of path? If it is not possible to determine if \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) is independent of path clearly explain why not.
- Evaluate \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) where \(f\left( {x,y,z} \right) = z{x^2} + x{\left( {y - 2} \right)^2}\) and \(C\) is the line segment from \(\left( {1,2,0} \right)\) to \(\left( { - 3,10,9} \right)\) followed by the line segment from \(\left( { - 3,10,9} \right)\) to \(\left( {6,0,2} \right)\).
- Evaluate \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) where \(f\left( {x,y} \right) = 4x + 3x{y^2} - \ln \left( {{x^2} + {y^2}} \right)\) and \(C\) is the upper half of \({x^2} + {y^2} = 1\) with clockwise rotation followed by the right half of \(\displaystyle {\left( {x - 1} \right)^2} + \frac{{{{\left( {y - 2} \right)}^2}}}{4} = 1\) with counter clockwise rotation. See the sketch below.
