Consider the curve below.
x = (cos(t))^2
y = cos(t)
0 ≤ t ≤ 6π
(a) Find the distance traveled by a particle with position (x, y) as t varies in the given time interval.
(b) What is the length of the curve?
I have no idea on how to evaluate this integral. I know that for part b you just divide the answer of a by 6 and that would give me the arc length. Any help would be great!
Is it just me overthinking it?
To find the distance traveled by a particle along the curve, we need to evaluate the integral of the speed of the particle with respect to time.
The speed can be calculated using the formula: speed = √((dx/dt)^2 + (dy/dt)^2)
Let's start by finding the derivatives dx/dt and dy/dt using the given parametric equations:
dx/dt = 2(cos(t))(−sin(t)) = −2sin(t)cos(t)
dy/dt = −sin(t)
Next, square both of the derivatives to get the speed formula:
(speed)^2 = (−2sin(t)cos(t))^2 + (−sin(t))^2
= 4(sin^2(t)cos^2(t)) + sin^2(t)
= 4sin^2(t)cos^2(t) + sin^2(t)
= sin^2(t)(4cos^2(t) + 1)
Now, integrate (speed)^2 with respect to t over the given time interval 0 ≤ t ≤ 6π:
∫[(speed)^2] dt = ∫[sin^2(t)(4cos^2(t) + 1)] dt
To evaluate this integral, we can use trigonometric identities. Rewrite sin^2(t) as (1 - cos^2(t)):
∫[(speed)^2] dt = ∫[(1 - cos^2(t))(4cos^2(t) + 1)] dt
Expanding the expression:
∫[4cos^2(t) + 1 - 4cos^4(t) - cos^2(t)] dt
Combine like terms:
∫[3cos^2(t) - 4cos^4(t) + 1] dt
Now you can integrate each term separately:
∫[3cos^2(t) dt] - ∫[4cos^4(t) dt] + ∫[1 dt]
The integral of cos^2(t) can be easily solved using a trigonometric identity:
∫[cos^2(t) dt] = 0.5(t + sin(t)cos(t))
The integral of cos^4(t) is a bit more involved. You can use the reduction formula to simplify it:
∫[cos^4(t) dt] = 0.375(t + 0.5sin(2t) + 0.125sin(4t))
The integral of 1 is simply t:
∫[1 dt] = t
Now substitute the values back into the original expression:
[3cos^2(t) - 4cos^4(t) + 1] t - 0.5(t + sin(t)cos(t)) - 0.375(t + 0.5sin(2t) + 0.125sin(4t))
Evaluate this expression for the upper limit (6π) and subtract the result of evaluating it for the lower limit (0):
[3cos^2(6π) - 4cos^4(6π) + 1] (6π) - 0.5[(6π) + sin(6π)cos(6π)] - 0.375[(6π) + 0.5sin(12π) + 0.125sin(24π)]
This will give you the distance traveled by the particle.
For part (b) of the question, you correctly mentioned that the length of the curve can be obtained by dividing the answer from part (a) by the length of the time interval. In this case, the length of the time interval is 6π since 0 ≤ t ≤ 6π. So, the length of the curve is the answer from part (a) divided by 6π.
nope. Look up arc length for parametric curves.
ds^2 = (dx/dt)^2 + (dy/dt)^2
The distance traveled is the arc length.