Final answer:
To find the length of the curve for parametric equations x = cos(t) and y = t × sin(t), differentiate both with respect to t, square and sum the derivatives, take the square root, and then integrate from 0 to π.
Step-by-step explanation:
You have been given the parametric equations x = cos(t) and y = t × sin(t), with the parameter t ranging from 0 to π (pi). To find the length of the curve described by these parametric equations, you need to evaluate the line integral over the given interval for t using the formula for the length of a curve in parametric form:
The length L of a smooth curve given by parametric equations x(t) and y(t) from t=a to t=b is defined by:
L = ∫ab √[ (Òx/Òt)² + (Òy/Òt)² ] dt
First, you differentiate x and y with respect to t to get Òx/Òt and Òy/Òt. Then, you square each of these derivatives and add them together under the square root. Afterward, integrate with respect to t from 0 to π to find the length of the curve.
The procedure includes the following steps:
Note that you will need to use integration techniques and trigonometric identities to evaluate the integral properly. Calculus and proficiency in integration are required to solve this type of problem.