Question

Find a length on a curve. ex: (t², t-t³/3) from -√3 to √3

a) Arc length

b) Curve integral

c) Surface area

d) Parametric equation

Answer 1

The question concerns the calculation of the arc length for a curve defined by parametric equations with a specific range for t. The arc length formula in parametrics requires integration of the root of sum of squares of the derivatives of x and y over the interval [-√3, √3].

Step-by-step explanation:

The question is asking for the calculation of the arc length of a curve represented by parametric equations x = t² and y = t - t³/3, with t ranging from -√3 to √3. To find the arc length, we need to calculate the integral of the square root of the sum of the squares of the derivatives of x and y with respect to t, over the given interval. The formula for the arc length (s) in parametric form is:

1. Compute dx/dt and dy/dt.
2. Find the square root of (dx/dt)² + (dy/dt)².
3. Integrate the result from the step above over the interval from -√3 to √3.

The other parts of the student's question, namely curve integral, surface area, and parametric equation, need additional context to provide a precise answer. However, the given parametric equations x = t² and y = t - t³/3 themselves represent the parametric equation of the curve.