For an object whose velocity in ft/sec is given by v(t) = cos(t), what is its distance, in feet, travelled on the interval t = 1 to t = 5?
Mathematics
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Question
For an object whose velocity in ft/sec is given by v(t) = cos(t), what is its distance, in feet, travelled on the interval t = 1 to t = 5?
1 Answer

1. User Answers LammettHash
The total distance traveled is given by
[tex]\displaystyle\int_1^5v(t)\,\mathrm dt[/tex]
You have
[tex]\displaystyle\int_1^5\cos t\,\mathrm dt=\int_1^{\pi/2}\cos t\,\mathrm dt\int_{\pi/2}^{3\pi/2}\cos t\,\mathrm dt+\int_{3\pi/2}^5\cos t\,\mathrm dt[/tex]
[tex]=\left(\sin\dfrac\pi2\sin1\right)\left(\sin\dfrac{3\pi}2\sin\dfrac\pi2\right)+\left(\sin5\sin\dfrac{3\pi}2\right)[/tex]
[tex]=(1\sin1)(11)+(\sin5(1))[/tex]
[tex]=4\sin1+\sin5[/tex]
[tex]\approx2.1996\text{ ft}[/tex]