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### Section 16.5 : Fundamental Theorem for Line Integrals

1. Evaluate $$\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$$ where $$f\left( {x,y} \right) = {x^3}\left( {3 - {y^2}} \right) + 4y$$ and C is given by $$\vec r\left( t \right) = \left\langle {3 - {t^2},5 - t} \right\rangle$$ with $$- 2 \le t \le 3$$. Solution
2. Evaluate $$\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$$ where $$f\left( {x,y} \right) = y{{\bf{e}}^{{x^{\,2}} - 1}} + 4x\sqrt y$$ and C is given by $$\vec r\left( t \right) = \left\langle {1 - t,2{t^2} - 2t} \right\rangle$$ with $$0 \le t \le 2$$. Solution
3. Given that $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ is independent of path compute $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where C is the ellipse given by $$\displaystyle \frac{{{{\left( {x - 5} \right)}^2}}}{4} + \frac{{{y^2}}}{9} = 1$$ with the counter clockwise rotation. Solution
4. Evaluate $$\displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}$$ where $$f\left( {x,y} \right) = {{\bf{e}}^{x\,y}} - {x^2} + {y^3}$$ and C is the curve shown below.
Solution