Consider the parametric equation?

Question

X= 10(cosx+xsinx)
Y= 10(sinx-xcosx)
What is the length of the curve from 0 to (3pi/2)

Answer 1

Length = INTEGRAL[0, 3π/2] sqrt(X'^2 + Y'^2) dx =
= INTEGRAL[0, 3π/2] sqrt((10x*cosx)^2 + (10x*sinx)'^2) dx =
= INTEGRAL[0, 3π/2] sqrt(10x*sqrt((cosx)^2 + (sinx)'^2) dx =
= INTEGRAL[0, 3π/2] 10x dx = 5x^2[x=0, x=3π/2] = 45π^2/4

Answer 2

If ds is an infinitesimal length along the curve then:
ds^2 = dY^2 + dX^2

dX = 10(-sinx dx + xcosx dx + sinx dx)
dX = 10x cosx dx

dY = 10(cosx dx - cosx dx + xsinx dx)
dY = 10x sinx dx

ds^2 = 100x^2( cos^2x + sin^2x) dx^2
ds^2 = 100x^2 dx^2
ds = 10x dx

Integrate to get:
10x^2/2 = 5x^2

Evaluate from 0 to 3pi/2:
5(3pi/2)^2 - 0 = 45pi^2/4

The length of the curve is 45pi^2/4