Consider the parametric equation: #x = 15(cos(theta) + (theta)sin(theta))# and #y = 15(sin(theta) - (theta)cos(theta))#, What is the length of the curve for #theta = 0# to #theta = pi/8#?

Question

Ezekiel Alexander · Accepted Answer

# (15pi^2)/128 #

The Arc Length for a Parametric Curve is given by

# L=int_alpha^beta sqrt((dx/dt)^2+(dy/dt)^2) \ dt #

So in this problem we have (using the product rule):

# \ \ \ \ \ \ \ \ x= 15(cos theta + theta sin theta) # # \ \ \ \ \ \ \ \ \ \ = 15 cos theta+15 theta sin theta # # :. dx/(d theta) = -15 sin theta + (15 theta)(cos theta) + (15)(sin theta)# # \ \ \ \ \ \ \ \ \ \ = 15 theta cos theta #

# \ \ \ \ \ \ \ \ y= 15(sin theta - theta cos theta) # # \ \ \ \ \ \ \ \ \ \ = 15 sin theta - 15 theta cos theta) # # :. dy/(d theta) = 15 cos theta - (15 theta)(sin theta) - (15)(cos theta) # # \ \ \ \ \ \ \ \ \ \ = - 15 theta sin theta #

So the Arc Length is;

# L= int_0^(pi/8) sqrt( (15 theta cos theta)^2 + (- 15 theta sin theta)^2) \ d theta # # \ \ = int_0^(pi/8) sqrt( 15^2 theta^2 cos^2 theta + 15^2 theta^2 sin^2 theta) \ d theta # # \ \ = int_0^(pi/8) sqrt( 15^2 theta^2 (cos^2 theta + sin^2 theta) \ d theta # # \ \ = int_0^(pi/8) sqrt( 15^2 theta^2 ) \ d theta # # \ \ = int_0^(pi/8) 15 theta \ d theta # # \ \ = [15/2 theta^2]_0^(pi/8) # # \ \ = 15/2 (pi^2/64-0) # # \ \ = (15pi^2)/128 #

Paisley Johnson · Answer

undefined To find the length of the curve defined by the parametric equations $ x = 15(\cos(\theta) + \theta\sin(\theta)) $ and $ y = 15(\sin(\theta) - \theta\cos(\theta)) $ for $ \theta = 0 $ to $ \theta = \frac{\pi}{8} $, follow these steps:

1. Compute the derivatives of $ x $ and $ y $ with respect to $ \theta $ to obtain expressions for $ \frac{dx}{d\theta} $ and $ \frac{dy}{d\theta} $.

2. Use the formula for the arc length of a parametric curve:
$$ L = \int_{\theta_1}^{\theta_2} \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} \, d\theta $$

3. Substitute the expressions for $ \frac{dx}{d\theta} $ and $ \frac{dy}{d\theta} $ into the formula.

4. Integrate the resulting expression from $ \theta = 0 $ to $ \theta = \frac{\pi}{8} $.

5. Evaluate the integral to find the length of the curve.