Given the velocity function \( v \) of an object moving along a line, explain how definite integrals can be used \( t 0 \) find the displacement of the object. Choose the correct answer below. A. The displacement between \( t=a \) and \( t=b,[a, b] \), is \( \int_{a}^{b} v(t) d t \). The displacement between \( t=a \) and \( t=b,[a, b] \), is \( \int_{a}^{b}|v(t)| d t \). C. The displacement between \( t=a \) and \( t=b,[a, b] \), is \( \int_{b}^{a} v(t) d t \). D. The displacement between \( t=a \) and \( t=b,[a, b] \), is \( \int_{b}^{a}|v(t)| d t \).

To determine the displacement of an object moving along a line using its velocity function \( v(t) \), we utilize definite integrals. The displacement over a time interval \([a, b]\) is given by the integral of the velocity function over that interval. Mathematically, this is expressed as: \[ \text{Displacement} = \int_{a}^{b} v(t) \, dt \] This integral calculates the net change in position from time \( t = a \) to \( t = b \). It's important to note that this gives the **displacement**, which is the overall change in position considering direction (i.e., it can be positive or negative). On the other hand, if you want to calculate the **total distance traveled** (regardless of direction), you would integrate the absolute value of the velocity function: \[ \text{Total Distance} = \int_{a}^{b} |v(t)| \, dt \] Given the options provided, **Option A** correctly states that the displacement is the integral of the velocity function over the interval \([a, b]\). However, it seems to have an incomplete or duplicated part in the statement. Ignoring the potential typographical error, **Option A** aligns with the correct mathematical interpretation for displacement. **Correct Answer:** **A. The displacement between \( t=a \) and \( t=b,[a, b] \), is \( \int_{a}^{b} v(t) d t \). The displacement between \( t=a \) and \( t=b,[a, b] \), is \( \int_{a}^{b}|v(t)| d t \).**