s = ∫[0,π] √((dx/dt)^2 + (dy/dt)^2) dt
OH, I think I messed up somewhere! I got 25.693, is this right? Please Help!
You know, it would be better to attach follow-ups to the actual question, where all the information is available.
I asked you for the actual integral you used. What did you actually evaluate?
Sorry, I will do that from now. s = ∫[0,π] √((dx/dt)^2 + (dy/dt)^2) dt I used my calculator and plugged in the equation and got 25.693, is this right? I can't really explain the steps to how I got this answer
I guess. Without actually seeing the values you used for dx/dt and dy/dt I have no way of knowing. If you relied on your calculator to determine those derivatives, then I certainly hope that was the right answer.
To find the value of the integral
s = ∫[0,π] √((dx/dt)² + (dy/dt)²) dt,
we need to evaluate the integral using the given limits of [0, π].
First, let's rewrite this integral as
s = ∫[0,π] √(dx/dt)² + (dy/dt)² dt.
Notice that the integrand inside the square root is the magnitude of the derivative of a vector function r(t) = (x(t), y(t)) with respect to t. This represents the instantaneous rate of change of the position vector r(t) as t varies.
To find the magnitude of the derivative of a vector function, we can use the chain rule for derivatives. Recall that
|d𝑟/dt| = √((dx/dt)² + (dy/dt)²).
Now, let's express the integral using the magnitude of the derivative:
s = ∫[0,π] |d𝑟/dt| dt.
The integrand, |d𝑟/dt|, represents the speed of the particle moving along the curve defined by the vector function r(t). Thus, the integral is actually calculating the total arc length along the curve between t = 0 and t = π.
To evaluate this integral, we need the explicit parametric equations for x(t) and y(t) that define the curve.
Once we have the parametric equations, we can find dx/dt and dy/dt by differentiating x(t) and y(t) with respect to t.
Finally, substitute these expressions for dx/dt and dy/dt into the formula for |d𝑟/dt| and integrate the resulting expression over the interval [0, π].
Without the specific form of x(t) and y(t), it is not possible to determine whether your answer of 25.693 is correct. I recommend double-checking your calculations and ensuring that you correctly evaluated the derivative and performed the integration.