The velocity of a particle moving along the x-axis is v(t) = cos(2t), with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance travelled by the particle from t = 0 to t = π minutes.
To find the total distance traveled by the particle from t = 0 to t = π minutes, we need to integrate the absolute value of the velocity function:
∫|v(t)| dt
First, let's find the integral of the absolute value function |v(t)|. Since v(t) = cos(2t), we have:
|v(t)| = |cos(2t)|
The absolute value of cos(2t) is always positive, so we can simplify the integral to:
∫cos(2t) dt
Next, we can find the antiderivative of cos(2t). The antiderivative of cos(x) is sin(x), so the antiderivative of cos(2t) is sin(2t). Thus, the integral becomes:
∫cos(2t) dt = (1/2)sin(2t) + C
Where C is the constant of integration.
Now, we can evaluate this antiderivative at the upper and lower limits of integration, t = π and t = 0:
(1/2)sin(2π) - (1/2)sin(0)
Since sin(2π) = sin(0) = 0, the expression simplifies to:
(1/2) * 0 - (1/2) * 0 = 0
Therefore, the total distance traveled by the particle from t = 0 to t = π minutes is 0 feet.
To find the total distance traveled by a particle, we need to calculate the definite integral of its velocity function over the given time interval.
Given that the velocity function of the particle is v(t) = cos(2t), we want to find the integral of this function from t = 0 to t = π.
To calculate the definite integral, follow these steps:
1. Find the antiderivative of the velocity function, which is obtained by integrating cos(2t) with respect to t. The antiderivative of cos(2t) is sin(2t)/2.
2. Evaluate the antiderivative at the upper limit of integration (π) and subtract the value evaluated at the lower limit of integration (0).
F(π) - F(0)
where F(t) is the antiderivative of v(t).
F(π) = sin(2π)/2 = 0
F(0) = sin(0)/2 = 0
3. The final step is to subtract the lower value from the upper value:
F(π) - F(0) = 0 - 0 = 0.
Therefore, the total distance traveled by the particle from t = 0 to t = π is 0 feet.
∫[0,π] cos(2t) dt = 0
How can this be? It is because the integral gives the displacement after π minutes, rather than the distance traveled. What you want to integrate is speed, rather than velocity.
∫[0,π] |cos(2t)| dt = 2
or, due to the symmetry of the graph, you could alo just take
4∫[0,π/4] cos(2t) dt = 2