Find the length of the curve correct to four decimal places. (Use a calculator to approximate the integral).
r(t) = (cos π t, 2t, sin 2πt), from (1, 0, 0) to (1, 4, 0)
This what I did.
r'(t)=-πsin(πt),2,2πcos(2πt)
lr'(t)l=√π^2sin^2(πt)+4+4π^2cos^2(2πt)
L=∫√π^2sin^2(πt)+4+4π^2cos^2(2πt)
I don't know integrate from what to what for ∫
Did I did any wrong in my steps above?
So far, so good. The curve from (1,0,0) to (1,4,0) is traced as t goes from 0 to 2. Better get out your calculator again.
Thank you
I don't get how you get 0,2. can you explain further ?
Well, let's see if we can make sense of this. Your calculation of the derivative of r(t), r'(t), seems to be correct. However, your calculation of the magnitude of r'(t), ||r'(t)||, seems a bit off.
To calculate the magnitude of r'(t), we have:
||r'(t)|| = √((π^2sin^2(πt)) + 4 + (4π^2cos^2(2πt)))
Now, let's proceed with finding the integral.
The length of a curve is given by the integral of the magnitude of the derivative of the curve. In this case, the curve is defined from t = 1 to t = 4.
So, we need to integrate ||r'(t)|| from t = 1 to t = 4:
L = ∫[1 to 4] √((π^2sin^2(πt)) + 4 + (4π^2cos^2(2πt))) dt
Unfortunately, this integral can't be expressed in terms of elementary functions, so we'll have to resort to numerical methods or calculations using a calculator or computer software.
Keep in mind that finding an approximation to this integral depends on the level of accuracy you need.
Your steps are correct up to finding the magnitude of r'(t). However, there is a minor mistake in finding the integral.
First, let's correct the magnitude of r'(t):
|r'(t)| = √[(-πsin(πt))^2 + 2^2 + (2πcos(2πt))^2]
= √[π^2sin^2(πt) + 4 + 4π^2cos^2(2πt)]
Now, to find the length of the curve, we need to integrate this magnitude from the initial value of t to the final value of t. In this case, the initial value is 1 and the final value is 4, as given in the question.
Therefore, the integral we need to evaluate is:
L = ∫[√(π^2sin^2(πt) + 4 + 4π^2cos^2(2πt))] dt, from 1 to 4
Now, to approximate the value using a calculator, you can use numerical methods such as the composite Simpson's rule or the trapezoidal rule. These methods involve dividing the interval [1, 4] into smaller sub-intervals and approximating the integral using a formula based on the function values at certain points within each sub-interval.
Alternatively, you can use software or calculators that have built-in integral functions, such as Wolfram Alpha or scientific graphing calculators, to directly evaluate the integral for you. Simply input the correct integrand and specify the limits of integration to obtain the numerical approximation of the length of the curve.
geez - the j component is just 2t, so as y goes from 0 to 4, t goes from 0 to 2.
cos and sin take care of themselves, too, right?
Your final integrand is a function of t, so you need to set limits of integration for t.