hi can you answer my question ? my question is 1-  calculate the arc length of the polar curve r= sin^3 (Q\3 ) from Q=0 TO Q=180\2

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to evaluate the arclength of the given curve, using polar coordinates formula such that:

`L = int_0^(pi/2)sqrt(r^2 + (dr/(d theta))^2) d theta`

`(dr/(d theta)) = (3sin^2(theta/3) cos(theta/3))/3`

`(dr/(d theta)) = sin^2(theta/3) cos(theta/3)`

`L = int_0^(pi/2)sqrt(sin^6(theta/3) + sin^2(theta/3) cos(theta/3)) d theta`

Factoring out `sin^2 (theta/3)` yields:

`L = int_0^(pi/2) sin (theta/3) sqrt(sin^4(theta/3) + cos(theta/3)) d theta`

`L = int_0^(pi/2) sin (theta/3) sqrt((1 - cos^2(theta/3))^2 + cos(theta/3)) d theta`

You need to come up with the substitution such that:

`cos (theta/3) = t => sin(theta/3) d theta = -3dt`

`cos 0 = 1`

`cos (pi/6) = sqrt3/2`

Changing the variable yields:

`L = int_1^(sqrt3/2) sqrt((1-t^2)^2 + t)(-3dt)`

`L = 3int_(sqrt3/2)^1 sqrt(1 - 2t^2 + t^4 + t) dt`

Hence, evaluating the arclength of the given curve yields `L = 3int_(sqrt3/2)^1 sqrt(1 - 2t^2 + t^4 + t) dt` .