Help would be appreciated, thank you! Bolded are my answers..
1. what is the length of the curve with parateric equtaions x=1-2cos(t), y=2-sin(t) from t=0 to tot=pi?
a. 8, b. 4, c. 8pi, d. 2pi
2. What is the length of the path along the graph of y=sqrt(9-x^2) between x=0 and x=2?
a. 3arcsin(1/3), b. 3arcsin(2/3), c. 2arcsin(2/3), d. 2arcsin(1/3)
3. what is the length of the curve x=3y^2/3(-1) from y=0 to y=8?
a. 16(sqrt 2), b. 16 (sqrt 2)-8, c. 16 (sqrt 2)+8, d. none of these
4. find the arc length of the curve from t=0 to t=2 whose derivatives in paramertic form are dx/dt cos(t) and dy/dt=ln(t+1). Write answer to two decimal places. Not sure about this one!!
5. Find the arc length on the interval for t between 0 and 1 inclusive for the curve described with the parametric equations x=1+3t^2, y=2t^3+4.
a. 2(sqrt 2)-1, b. 4 (sqrt 2)-2, c. 2(sqrt 2)-2, d. (sqrt 2)-2
What is the length of the path along the graph of y=sqrt(9-x^2) between x=0 and x=2?
dy/dx = (1/2) (9 - x^2)^(-1/2) ( -2x) = -x / ( 9 - x^2)^(1/2)
[dy/dx]^2 = x^2 / [ 9 - x^2]
The arc length is given by
2
∫ √ ( 1 + [dy/dx]^2 ) dx =
0
2
∫ √ ( 1 + x^2 / [ 9 - x^2] ) dx =
0
2
∫ √ ( 9 / [ 9 - x^2] ) dx =
0
2
3 ∫ √ ( 1 / [ 9 - x^2] ) dx =
0
let x = 3sin (θ) → x^2 = 9sin^2( θ)
dx = 3cos (θ) dθ
Change the limits of integration
when x = 2 when x = 0
2/3 = sin (θ) 0 = sin (θ)
θ = arcsin(2/3) θ = 0
So we have
arcsin(2/3)
3 ∫ √ ( 1 / [ 9 - 9sin^2( θ) ] ) 3 cos ( θ) dθ =
0
arcsin(2/3)
3 ∫ (1/3) √ ( 1 / [ 1 - sin^2( θ) ] ) 3 cos ( θ) dθ =
0
arcsin(2/3)
3 ∫ √ ( 1 / [ 1 - sin^2( θ) ] ) cos ( θ) dθ =
0
arcsin(2/3)
3 ∫ √ ( 1 / [cos^2 ( θ) ] ) cos ( θ) dθ =
0
arcsin(2/3)
3 ∫ ( 1 / [cos ( θ) ] ) cos ( θ) dθ =
0
arcsin(2/3)
3 ∫ dθ =
0
arcsin(2/3)
3 [ θ ] = 3 [ arcsin(2/3) - 0 ] = 3 arcsin(2/3)
0
Answer "b"
1. what is the length of the curve with parateric equtaions x=1-2cos(t), y=2-sin(t) from t=0 to tot=pi?
dx/dt = 2sin(t) .... [ dx/dt]^2 = 4sin^2(t)
dy/dt = -cos(t)..... [dy/dt]^2 = cos^2(t)
The arc length is found as
pi
∫ √ ( [ dx/dt]^2 + [ dy/dt]^2 ) dt =
0
pi
∫ √ ( 4sin^2(t) + cos^2 (t) ) dt
0
pi
∫ √ ( 4sin^2(t) + 1 - sin^2(t) ) dt
0
pi
∫ √ ( 3sin^2(t) + 1 ) dt ≈ 4.84422
0