Final answer:
To find the length of the curve from point A to point B, we can use the formula for the arc length of a curve. In this case, the curve is a circular arc with a radius of 2m, centered at (0, 2m). The arc length integral is given by L = ∫√(dx/dt)^2 + (dy/dt)^2 dt, where dx/dt and dy/dt are the derivatives of x and y with respect to t, respectively.
Step-by-step explanation:
To find the length of the curve from point A to point B, we can use the formula for the arc length of a curve. In this case, the curve is a circular arc with a radius of 2m, centered at (0, 2m). We can represent the curve as a parameterized function r(t) = (2cos(t), 2 + 2sin(t)), where t ranges from 0 to pi.
The arc length integral is given by L = ∫√(dx/dt)^2 + (dy/dt)^2 dt, where dx/dt and dy/dt are the derivatives of x and y with respect to t, respectively. In this case, dx/dt = -2sin(t) and dy/dt = 2cos(t).
Substituting these values into the arc length integral and integrating from 0 to pi, we get:
L = ∫√((-2sin(t))^2 + (2cos(t))^2) dt = ∫√(4sin^2(t) + 4cos^2(t)) dt = ∫2 dt = 2t
Evaluating the integral from 0 to pi, we have:
L = 2π - 0 = 2π