A particle is moving so that its velocity, v(t) = t-64/t^2 for 2≤t≤6. find the total distance traveled by the particle.
To find the total distance traveled by the particle, we need to integrate the absolute value of the velocity function over the interval [2, 6]. This is because distance is defined as the absolute value of displacement, which can be negative or positive.
Let's break down the problem into steps:
Step 1: Determine the interval of integration.
Given that the particle is moving from t = 2 to t = 6, we need to integrate the velocity function over this interval.
Step 2: Calculate the absolute value of the velocity function.
In this case, the velocity function is v(t) = t - 64/t^2. We need to take the absolute value of this function, which means disregarding the negative sign if it's present.
|v(t)| = |t - 64/t^2|
Step 3: Evaluate the integral.
To find the total distance traveled, we integrate the absolute value of the velocity function:
d = ∫[2, 6] |v(t)| dt
Integrating |v(t)| = |t - 64/t^2| over the interval [2, 6] will give us the total distance traveled by the particle.
Unfortunately, the integration process can't be straightforward due to the complexity of the function. To calculate the integral, you can use numerical methods like Simpson's rule, Trapezoidal rule, or numerical integration software. These methods approximate the integral and give you an estimated result.
Alternatively, if the function has a simpler form, you could try to simplify or rearrange the equation to make it easier to integrate.
Once you have obtained the result of the integral, that will be the total distance traveled by the particle.