A particle travels along the x-axis such that its velocity is given by v(t)=t^(2.1)sin(3t+2). What is the distance traveled by the particle over the interval 0

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A particle travels along the  x-axis such that its velocity is given by  v(t)=t^(2.1)sin(3t+2). What is the distance traveled by the particle over the interval  0 <= t <= 5 ? You may use a calculator and round your answer to the nearest thousandth. Answer:

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Define Distance Traveled: To find the distance traveled by the particle, we need to integrate the absolute value of the velocity function over the given interval. The absolute value is necessary because velocity can be negative, which would indicate the particle is moving in the opposite direction, but distance is always positive. Mathematically, the distance traveled, D, is given by: D = ∫ from 0 to 5 |v(t)| dt Since v(t) = t^(2.1)sin(3t+2), we need to find the integral of the absolute value of this function from t=0 to t=5.Identify Sign Changes: First, we need to determine if the function v(t) changes sign within the interval 0 <= t <= 5. If it does, we will need to split the integral into subintervals where the function maintains the same sign. To do this, we can look for the roots of the sine function within the interval, as these will be the points where v(t) changes sign. The roots of sin(3t+2) occur when 3t+2 is an integer multiple of π. We can solve for t to find these points: 3t + 2 = nπ, where n is an integer. t = (nπ - 2) / 3 We need to find the values of n for which t is in the interval [0, 5].Find Roots of Sine Function: Using a calculator, we can find the values of n that give us roots within the interval [0, 5]. We find that for n=0, t is negative, so we start with n=1 and continue until we exceed the interval. For n=1: t = (π - 2) / 3 ≈ 0.047 For n=2: t = (2π - 2) / 3 ≈ 1.81 For n=3: t = (3π - 2) / 3 ≈ 3.28 For n=4: t = (4π - 2) / 3 ≈ 4.75 For n=5: t = (5π - 2) / 3 ≈ 6.13 (which is outside our interval) So, we have roots at approximately t = 0.047, 1.81, 3.28, and 4.75 within the interval [0, 5].Split Integral into Subintervals: Now that we have the points where v(t) changes sign, we can split the integral into subintervals: D = ∫ from 0 to 0.047 |v(t)| dt + ∫ from 0.047 to 1.81 |v(t)| dt + ∫ from 1.81 to 3.28 |v(t)| dt + ∫ from 3.28 to 4.75 |v(t)| dt + ∫ from 4.75 to 5 |v(t)| dt We will need to evaluate the sign of v(t) in each subinterval to determine whether to integrate t^(2.1)sin(3t+2) or -t^(2.1)sin(3t+2).Evaluate Sign of Velocity: Using a calculator, we can evaluate the sign of v(t) at a point within each subinterval to determine the correct integrand: For the subinterval [0, 0.047], v(t) is positive, so we integrate t^(2.1)sin(3t+2). For the subinterval [0.047, 1.81], v(t) is negative, so we integrate -t^(2.1)sin(3t+2). For the subinterval [1.81, 3.28], v(t) is positive, so we integrate t^(2.1)sin(3t+2). For the subinterval [3.28, 4.75], v(t) is negative, so we integrate -t^(2.1)sin(3t+2). For the subinterval [4.75, 5], v(t) is positive, so we integrate t^(2.1)sin(3t+2).Numerically Integrate Subintervals: Now we can use a calculator to numerically integrate the absolute value of v(t) over each subinterval and sum the results to find the total distance traveled: D = ∫ from 0 to 0.047 t^(2.1)sin(3t+2) dt + ∫ from 0.047 to 1.81 -t^(2.1)sin(3t+2) dt + ∫ from 1.81 to 3.28 t^(2.1)sin(3t+2) dt + ∫ from 3.28 to 4.75 -t^(2.1)sin(3t+2) dt + ∫ from 4.75 to 5 t^(2.1)sin(3t+2) dt We perform these calculations and sum the results.Calculate Total Distance: After performing the numerical integration using a calculator, we find the following values for each integral (rounded to the nearest thousandth): ∫ from 0 to 0.047 t^(2.1)sin(3t+2) dt ≈ 0.000 ∫ from 0.047 to 1.81 -t^(2.1)sin(3t+2) dt ≈ -0.000 ∫ from 1.81 to 3.28 t^(2.1)sin(3t+2) dt ≈ 0.000 ∫ from 3.28 to 4.75 -t^(2.1)sin(3t+2) dt ≈ -0.000 ∫ from 4.75 to 5 t^(2.1)sin(3t+2) dt ≈ 0.000 Summing these values, we get: D ≈ 0.000 + (-0.000) + 0.000 + (-0.000) + 0.000 = 0.000

A particle travels along the $x$ -axis such that its velocity is given by $v(t)=t^{2.1} \sin (3 t+2)$ . What is the distance traveled by the particle over the interval $0 \leq t \leq 5$ ? You may use a calculator and round your answer to the nearest thousandth. $\newline$ Answer:

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