A particle moves along a line with a velocity v(t)=t2+3t−4, measured in meters per second. Find the total distance the particle travels from t=0 seconds to t=2 seconds.

To find the total distance traveled by the particle, we need to find the integral of the absolute value of the velocity function from 0 to 2. However, since the velocity function is a polynomial, it is easier to find the areas under the curve from 0 to 2 and add them up.

First, we need to find the roots of the velocity function v(t) = t^2 + 3t - 4. This is done by setting the function equal to zero and solving for t:

t^2 + 3t - 4 = 0

This can be factored into:

(t - 1)(t + 4) = 0

Setting each factor equal to zero gives the roots t = 1 and t = -4. Since we are only interested in the interval from 0 to 2, we only consider the root t = 1.

This means that the velocity function changes sign at t = 1. Therefore, we need to split the integral into two parts: from 0 to 1, and from 1 to 2.

The total distance traveled by the particle from t = 0 to t = 2 is then given by:

∫ from 0 to 1 (t^2 + 3t - 4) dt + ∫ from 1 to 2 -(t^2 + 3t - 4) dt

The first integral represents the distance traveled from t = 0 to t = 1, and the second integral represents the distance traveled from t = 1 to t = 2. The negative sign in the second integral is because the velocity is negative in this interval, indicating that the particle is moving in the opposite direction.

By calculating these integrals, we can find the total distance traveled by the particle from t = 0 to t = 2.