What is the arclength of r=-10sin(theta/4+(5pi)/16) on theta in [(-5pi)/16,(9pi)/16]?

1 Answer
May 27, 2018

L=(133pi)/20-48/5sin(7/16pi)cos(11/8pi)+40sum_(n=2)^oo((1/2),(n))(-6/25)^n{((2n),(n))(7pi)/32+2sum_(k=0)^(n-1)((2n),(k))(sin((n-k)(7pi)/16)cos((n-k)(11pi)/8))/(n-k)} units.

Explanation:

r=−10sin(θ/4+(5pi)/16)
r^2=100sin^2(θ/4+(5pi)/16)

r'=−10/4cos(θ/4+(5pi)/16)
(r')^2=100/25cos^2(θ/4+(5pi)/16)

Arclength is given by:

L=int_((-5pi)/16)^((9pi)/16)sqrt(100sin^2(θ/4+(5pi)/16)+100/25cos^2(θ/4+(5pi)/16))d theta

Apply the substitution θ/4+(5pi)/16=phi and rearrange:

L=40int_((15pi)/64)^((29pi)/64)sqrt(1-24/25cos^2phi)dphi

For phi in [(15pi)/64,(29pi)/64], 24/25cos^2phi<1. Take the series expansion of the square root:

L=40int_((15pi)/64)^((29pi)/64)sum_(n=0)^oo((1/2),(n))(-24/25cos^2phi)^ndphi

Isolate the n=0 term and rearrange:

L=40int_((15pi)/64)^((29pi)/64)dphi+40sum_(n=1)^oo((1/2),(n))(-24/25)^nint_((15pi)/64)^((29pi)/64)cos^(2n)phidphi

Apply the trigonometric power-reduction formula:

L=(35pi)/4+40sum_(n=1)^oo((1/2),(n))(-24/25)^nint_((15pi)/64)^((29pi)/64){1/4^n((2n),(n))+2/4^nsum_(k=0)^(n-1)((2n),(k))cos((2n-2k)phi)}dphi

Integrate term by term:

L=(35pi)/4+40sum_(n=1)^oo((1/2),(n))(-6/25)^n[((2n),(n))phi+sum_(k=0)^(n-1)((2n),(k))sin((2n-2k)phi)/(n-k)]_((15pi)/64)^((29pi)/64)

Insert the limits of integration:

L=(35pi)/4+40sum_(n=1)^oo((1/2),(n))(-6/25)^n{((2n),(n))(7pi)/32+sum_(k=0)^(n-1)((2n),(k))(sin((n-k)(29pi)/32)-sin((n-k)(15pi)/32))/(n-k)}

Apply the trigonometric sum-to-product formula:

L=(35pi)/4+40sum_(n=1)^oo((1/2),(n))(-6/25)^n{((2n),(n))(7pi)/32+2sum_(k=0)^(n-1)((2n),(k))(sin((n-k)(7pi)/16)cos((n-k)(11pi)/8))/(n-k)}

Isolate the n=1 terms:

L=(35pi)/4-24/5((7pi)/16+2sin(7/16pi)cos(11/8pi))+40sum_(n=2)^oo((1/2),(n))(-6/25)^n{((2n),(n))(7pi)/32+2sum_(k=0)^(n-1)((2n),(k))(sin((n-k)(7pi)/16)cos((n-k)(11pi)/8))/(n-k)}

Simplify:

L=(133pi)/20-48/5sin(7/16pi)cos(11/8pi)+40sum_(n=2)^oo((1/2),(n))(-6/25)^n{((2n),(n))(7pi)/32+2sum_(k=0)^(n-1)((2n),(k))(sin((n-k)(7pi)/16)cos((n-k)(11pi)/8))/(n-k)}