\( \left. \begin{array} { c } { \lim _ { n \rightarrow \infty } \sum _ { j = 1 } ^ { n } \frac { 16 } { n } f ( 6 + \frac { 4 j } { n } ) = 5 } \\ { \int _ { 6 } ^ { 10 } f ( x ) d x = \square } \end{array} \right. \)

Label each expression as a scalar quantity, a vector quantity or undefined, if \( f \) is a scalar function and \( F \) is a vector field. \( \nabla \times(\nabla f) \) scalar vector

52. \( \star \sum_{n=1}^{\infty} \sqrt{n}(\sin n)\left(\sin \frac{1}{n}\right)^{2} \)

\( \sum _ { y = 1 } ^ { n } \sum _ { x = 1 } ^ { n } 4 x + 7 y + x ^ { 2 } \)

Compute the work done by the force field \( F \) along the curve \( C \). \( \mathrm{F}(x, y)=\langle 2 x, 2 y\rangle, C \) is the line segment from \( (1,1) \) to \( (5,4) \) \( W=\square \)

Evaluate the line integral \( \int_{C} 5 x d s \), where \( C \) is the line segment from \( (1,2) \) to \( (8,7) \). \( \int_{C} 5 x d s=\square \)