Question

Sketch the plane curve and find its length over the given interval.

Answer 1

Step 1:

The curve is .

Compare the curve with .

Here , .

Graph:

Grah the parametric equations  in the interval [0, 1].

Step 2:

Arc length:

If the curve has the vector equation , then arc length of the curve is defined as,

, where

Differentiate with respect to on each side .

Differentiate with respect to on each side .

Substitute values in .

Step 3:

Substitute and limits of the integral in the arc length formula.

Let .

Differentiate with respect to on each side .

Change of integral limits:

Upper limit: If , then .

Lower limit: If , then .

Substitute  , and limits of integral limits in the integral.

Solution:

Arc length of the curve is .