Question

Using the differential length dl, find the length of each of the following curves:

(a) p = 3, pi/4 < theta < pi/2, z = constant
(b) T = 1, theta = 30°, 0 < theta < 60°
(c) r = 4, 30º < theta < 90° theta = constant

Answer 1

To find the length of each curve, use the concept of arc length and the formula s = ∫√(r² + (dr/dθ)²) dθ. Substitute the given values into the formula and integrate to find the length.

Step-by-step explanation:

To find the length of each curve, we need to use the concept of arc length. The formula for the arc length of a curve given in polar coordinates is s = ∫√(r² + (dr/dθ)²) dθ, where r is the radius and dr/dθ is the derivative of r with respect to θ.

(a) For p = 3, pi/4 < θ < pi/2, z = constant, we need to find the length of the curve. Since the value of p is constant, we can substitute p = 3 into the formula and integrate to find the length.

(b) For T = 1, θ = 30°, 0 < θ < 60°, we substitute T = 1 into the formula and integrate to find the length of the curve.

(c) For r = 4, 30º < θ < 90°, θ = constant, we substitute r = 4 into the formula and integrate to find the length of the curve.