# The force applied against an object moving horizontally on a linear path is described by F(x)= cospix+3x . By how much does the object's kinetic energy change as the object moves from  x in [ 1, 2 ]?

Feb 9, 2016

$\setminus \Delta K = W = \setminus {\int}_{1}^{2} F \left(x\right) \mathrm{dx} = \frac{9}{2}$ Joules

#### Explanation:

Work-Energy Theorem: The total work done by all the forces acting on a body must be equal to the change in its kinetic energy.

$\setminus \Delta K = {K}_{f} - {K}_{i} = W$

$W \setminus \equiv \setminus {\int}_{1}^{2} F \left(x\right) \mathrm{dx} = \setminus {\int}_{1}^{2} \left(\cos \setminus \pi x + 3 x\right) \mathrm{dx}$,

$\setminus q \quad$ $= \frac{1}{\setminus} \pi \setminus {\int}_{1}^{2} \setminus \frac{d}{\mathrm{dx}} \left(\sin \setminus \pi x\right) \mathrm{dx} + {\left[\frac{3}{2} {x}^{2}\right]}_{1}^{2}$

$\setminus q \quad$ $= \setminus \frac{1}{\pi} {\left[\sin \setminus \pi x\right]}_{1}^{2} + \frac{9}{2} = 0 + \frac{9}{2} = 4.5 J$