Line Integrals:

1. Evaluate \( \displaystyle{ \int_C (x^2y-1) \, ds}\), where \(C\) is the upper half of the circle \(x^2+y^2=4\).   solution

2. Evaluate \( \displaystyle{ \int_C \sqrt{2} xy \, ds}\), where \(C\) consists of the line segments from \( (0, 0) \) to \((2, 0)\), and from \((2, 0)\) to \((3, 1)\).  solution

3. Evaluate \( \displaystyle{ \int_C x^2 \, dx + y^2 \, dy}\), where \(C\) consists of the arc of the circle \(x^2+y^2=4\) from \((2, 0)\) to \((0,2)\) followed by the line segment \((0, 2)\) to \((4, 4)\).   solution

4. Evaluate \(\displaystyle{ \int_C 4\sqrt{13} xyz \, ds}\), where \(C\) is the part of the helix defined by \(x=2t, \: y=3\sin t, \: z=3\cos t,\) for \(0\le t \le \pi/2 \).  solution

5. Compute the line integral \( \displaystyle{ \int_C {\bf F} \cdot d{\bf r} }\) where  \({\bf F}(x,y,z) = x^2\,{\bf i} +xy\,{\bf j}+z^2 \,{\bf k}\)  and \(C\) is given by  \({\bf r}(t) = \cos t \, {\bf i} + \sin t \, {\bf j}+ t^2 \, {\bf k}\) for \( 0 \le t \le 1\).   solution

6. Compute the line integral \( \displaystyle{ \int_C {\bf F} \cdot d{\bf r}} \) where  \({\bf F}(x,y,z)=\langle z^2, x^2, y^2 \rangle\) and \(C\) is the line segment from \((1, 0, 0)\) to \( (4,1,2)\).   solution