3) Determine la longitud exacta de las siguientes curvas a) \( f(x)=1+6 x^{3 / 2}, \quad 0 \leqslant x \leqslant 1 \) b) \( f(x)=\frac{x^{3}}{3}+\frac{1}{4 x}, \quad 1 \leqslant x \leqslant 2 \)

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

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

Compute the work done by the force field \( F=\langle y, 0, z\rangle \) along the triangle from \( (0,0,0) \) to \( (2,3,4) \) then to \( (2,3,0) \) and then back to \( (0,0,0) \). \( W=\square \)

Evaluate the line integral. \( \int_{C} 5 z d s \), where \( C \) is the line segment from \( (5,0,1) \) to \( (6,-6,6) \) \( \int_{C} 5 z d s=\square \)

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 \( (5,4) \) \( W=\square \)