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?

Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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