The components of a position vector of a particle moving in the plane are components t cubed, 2 times the sine of t. What is the distance traveled by the particle from t = 1 to t = π?

Type your answer in the space below and give 3 decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.482).
Please can you show me your work? Thanks

Actually I have just done the test and got 100%

The answer is 30.113

first you do derivative of the vectors both coordinates.
Then you find the magnitude of the vector. and that magnitude is the derivative of the position function. After that you know what to do.
Just in your calc enter integral from 1 to pi of sqrt((etc)^2) and you get 30.113

ds^2 = dx^2 + dy^2

x = t^3
so dx = 3 t^2 dt and dx^2 = 9 t^6 dt^2

y = 2 sin t
dy = 2 cos t dt and dy^2 = 4 cos^2 t dt^2

ds^2 = dt^2 [ 9 t^6 + 4 cos^2 t ]
ds = [ 9t^2+4cos^2t]^2 dt
ds = [81 t^4 + 72 t^2 cos^2 t + 16 cos^4 t ] dt
s = [ (81/5)t^5 + 72(t^3/6+{t^2/4-1/8}sin 2t+tcos2t /4)+16 (sint cos^3t/4 +{3/8}costsint +3t/8) ] ... see https://socratic.org/questions/how-do-you-find-the-integral-of-cos-4-x-dx
evaluate at t = pi and t = 1 and subtract

sorry, to the 1/2 not squared, same approach

Omg!!! I am so confuse. I guess are the parenthesis or exponents. I don't know. The example in socratic is so different. I don't get it. Please I need your help

To find the distance traveled by the particle from t = 1 to t = π, we need to integrate the magnitude of the particle's velocity vector over that time interval.

The position vector of the particle is given as r(t) = (t^3, 2sin(t)), where t is the parameter representing time.

The velocity vector v(t) is the derivative of the position vector with respect to time:
v(t) = (d/dt(t^3), d/dt(2sin(t))) = (3t^2, 2cos(t))

The speed of the particle at any time t is the magnitude of the velocity vector, which is given by:
|v(t)| = √(3t^2)^2 + (2cos(t))^2 = √(9t^4 + 4cos^2(t))

To find the distance traveled from t = 1 to t = π, we integrate the speed function |v(t)| over that interval:
distance = ∫[1, π] √(9t^4 + 4cos^2(t)) dt

Unfortunately, this integral does not have an elementary solution, which means we cannot find a closed-form expression for the exact distance traveled. However, we can estimate the distance using numerical methods.

One way to approximate the distance is by using numerical integration techniques, such as the trapezoidal rule or Simpson's rule. These methods involve dividing the interval into smaller sub-intervals and approximating the integral using the function values at those points.

Alternatively, we can use a graphing calculator, software, or programming language that supports numerical integration, such as Python or MATLAB, to evaluate the integral and obtain an approximate value for the distance. Here is an example code using Python:

```python
import numpy as np
from scipy.integrate import quad

# Define the speed function
def speed(t):
return np.sqrt(9 * t**4 + 4 * np.cos(t)**2)

# Integrate the speed function over the interval [1,π]
distance, error = quad(speed, 1, np.pi)

# Print the approximate distance with 3 decimal places
print(f"Distance traveled: {distance:.3f}")
```

Running this code will give you an approximate value for the distance traveled by the particle from t = 1 to t = π.