Find the curvature of the curve with parametric equations So

Find the curvature of the curve with parametric equations:

Solution

Given x(t) = ?(0 to t) sin(p?²/2) d?, y(t) = ?(0 to t) cos(p?²/2) d?: x\'(t) = sin(pt²/2), y\'(t) = cos(pt²/2), by the Fundamental Theorem of Calculus x\'\'(t) = pt cos(pt²/2), y\'(t) = - pt sin(pt²/2). So, k = |x\' y\'\' - y\' x\'\'| / ((x\')² + (y\')²)^(3/2) = |sin(pt²/2) * -pt sin(pt²/2) - cos(pt²/2) * pt cos(pt²/2)| / 1^(3/2) = |-pt [sin²(pt²/2) + cos²(pt²/2)]| = pt.