Question

# A particle with velocity at any time t is given by v(t)=2e^{2t} moves in a straight line. How far does the particle travel during the time interval when its velocity increases from 2 to 4?

Integrals
A particle with velocity at any time t is given by $$\displaystyle{v}{\left({t}\right)}={2}{e}^{{{2}{t}}}$$ moves in a straight line. How far does the particle travel during the time interval when its velocity increases from 2 to 4?

2021-03-08
Given that the velocity of a particle at any time t is given by
$$\displaystyle{v}{\left({t}\right)}={2}{e}^{{{2}{t}}}$$
and it is moving in the straight line we have to find the displacement. Integrate the above Equation between the limits 2 to 4
Then
$$\displaystyle{s}={\int_{{2}}^{{4}}}{2}{e}^{{{2}{t}}}{\left.{d}{t}\right.}$$
$$\displaystyle={2}{{\left[{\frac{{{e}^{{{2}{t}}}}}{{{2}}}}\right]}_{{2}}^{{4}}}$$
$$\displaystyle={e}^{{8}}-{e}^{{4}}$$