# calculus ?

Find the distance traveled by a particle with position (x, y) as t varies in the given time interval.

x = cos^2(t), y = cos(t), 0 ≤ t ≤ 3𝜋

Update:

I know i have to use the formula

sqrt((dx/dt)^2+(dy/dt)^2)

Relevance
• x = cos(t)^2

dx/dt = 2 * cos(t) * (-sin(t)) = -2sin(t)cos(t)

y = cos(t)

dy/dt = -sin(t)

sqrt((dx/dt)^2 + (dy/dt)^2) * dt =>

sqrt(4 * sin(t)^2 * cos(t)^2 + sin(t)^2) * dt =>

sqrt(sin(t)^2 * (4cos(t)^2 + 1)) * dt =>

sin(t) * sqrt(4 * cos(t)^2 + 1) * dt

u = cos(t)

du = -sin(t) * dt

sqrt(4u^2 + 1) * (-du) =>

-sqrt(1 + 4u^2) * du

tan(m) = 2u

sec(m)^2 * dm = 2 * du

-sqrt(1 + tan(m)^2) * (1/2) * sec(m)^2 * dm =>

(-1/2) * sqrt(sec(m)^2) * sec(m)^2 * dm =>

(-1/2) * sec(m) * sec(m)^2 * dm

int(sec(m)^3 * dm)

a = sec(m)

da = sec(m) * tan(m) * dm

db = sec(m)^2 * dm

b = tan(m)

int(sec(m)^3 * dm) = sec(m) * tan(m) - int(sec(m) * tan(m)^2 * dm)

int(sec(m)^3 * dm) = sec(m) * tan(m) - int(sec(m) * (sec(m)^2 - 1) * dm)

int(sec(m)^3 * dm) = sec(m) * tan(m) - int(sec(m)^3 * dm) + int(sec(m) * dm)

2 * int(sec(m)^3 * dm) = sec(m) * tan(m) + int(sec(m) * dm)

2 * int(sec(m)^3 * dm) = sec(m) * tan(m) + ln|sec(m) + tan(m)|

int(sec(m)^3 * dm) = (1/2) * (sec(m) * tan(m) + ln|sec(m) + tan(m)|)

(-1/4) * (sec(m) * tan(m) + ln|sec(m) + tan(m)|) =>

(-1/4) * (2u * sqrt(1 + 4u^2) + ln|2u + sqrt(1 + 4u^2)|) =>

(-1/4) * (2 * cos(t) * sqrt(1 + 4 * cos(t)^2) + ln|2 * cos(t) + sqrt(1 + 4 * cos(t)^2)|)

From t = 0 to t = 3pi

cos(0) = 1

cos(3pi) = -1

(-1/4) * (2 * (-1) * sqrt(1 + 4 * 1) + ln|2 * (-1) + sqrt(1 + 4 * 1)|) + (1/4) * (2 * 1 * sqrt(1 + 4 * 1) + ln|2 * 1 + sqrt(1 + 4 * 1)|) =>

(1/4) * (2 * sqrt(5) + ln|2 + sqrt(5)|) - (1/4) * (-2 * sqrt(5) + ln|-2 + sqrt(5)|) =>

(1/4) * (2 * sqrt(5) + ln|2 + sqrt(5)|) + (1/4) * 2 * sqrt(5) - (1/4) * ln|sqrt(5) - 2|) =>

(1/4) * (2 * sqrt(5) + 2 * sqrt(5)) + (1/4) * ln|(sqrt(5) + 2) / (sqrt(5) - 2)| =>

(1/4) * 4 * sqrt(5) + (1/4) * ln|(sqrt(5) + 2)^2 / (5 - 4)| =>

sqrt(5) + ln|(5 + 4 * sqrt(5) + 4) / 1| =>

sqrt(5) + ln|9 + 4 * sqrt(5)| =>

sqrt(5) + ln(9 + 4 * sqrt(5))

• Hint: What formula or concept did you see for "the distance", or the "length of the curve" ?

State it here and we will show you how to apply it.