Chapter 5.1, Problem 5E

To determine

To find: The value of particle at $t=4$ .

Answer to Problem 5E

The answer is $x=25$ .

Explanation of Solution

Given information:

The total distance the particle travels is equal to the area under the curve $v(t)=t^2+1$

From $t=0$ to

  $t=4$ .

Calculation:

The time range $\left[0,4\right]$ should be divided into subintervals with lengths of

  $\text{Δ}t=1$ .

Approximate the area under the curve by finding the area of each rectangle whose base is $\text{Δ}t=1$ and the function's height is its $y$ -coordinate at the centre of its base.

  

Since the sum of the areas in the table above approximates the area under the curve $v(t)$ from $t\in [0,4]$

Area:

  $\frac{5}{4}+\frac{13}{4}+\frac{29}{4}+\frac{53}{4}=25$

Hence the particle start at $x=0$ at $t=4$

The particle will be:

  $\begin{array}{c}x=0+25\\ =25\end{array}$

Therefore the required particle at $t=4$ is $x=25$ .