Question

A particle moves along with a velocity of v(t) = 2t + 3
Find the total distance traveled from t=2 to t=5

Answer 1

$$
\large
\int_2^5v(t)dt\\
\int_2^52t+3~dt\\
=\left (\cfrac{2t^2}{2}+3t\right)\Huge{|^5_2}\\
=(25+15)-(4+6)=?~
$$

Answer 2

Do you know why this works?

Answer 3

so its 30? i think so.

Answer 4

It works because, velocity = distance traveled / time taken.
So,
$$
\int v(t)~dt=\int \cfrac{dx}{dt}dt=\int dx
$$
Which is just summing the infinitesimal distances, \(dx\) from 2 to 5.

Answer 5

thanks!

Answer 6

you're welcome