Answer:
4
Step-by-step explanation:
I used a calculator. I put integral values of 3 and 1 and then entered the equation inside absolute value bars
The total distance traveled by the laser beam in the given time interval is required.
The total distance traveled by the laser in the given time interval is 4 feet.
The given equation is
[tex]v(t)=t^2-4[/tex]
The given time interval is from t = 1 to t = 3 seconds.
In order to find the distance traveled we need to integrate the equation with respect to time and between the given intervals.
[tex]0=t^2-4\\\Rightarrow t=\pm 2=+2[/tex]
The laser will change the direction at [tex]t=2[/tex]
[tex]\int_1^3 v(t)=\int_1^2t^2-4 dt+\int_2^3t^2-4 dt\\\Rightarrow \int_1^3 v(t)=\dfrac{t^3}{3}-4t|_1^2+\dfrac{t^3}{3}-4t|_2^3\\\Rightarrow \int_1^3 v(t)=\left|\dfrac{2^3}{3}-4\times 2-(\dfrac{1^3}{3}-4)+\dfrac{3^3}{3}\right|-\left|4\times 3-(\dfrac{2^3}{3}-4\times2)\right|\\\Rightarrow \int_1^3 v(t)=\dfrac{5}{3}+\dfrac{7}{3}\\\Rightarrow \int_1^3 v(t)=\dfrac{12}{3}\\\Rightarrow \int_1^3 v(t)=4\ \text{m}[/tex]
Since, here distance is asked the absolute value of each integral is used.
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