Section 9.3

Area with Parametric Curves

We recall . By substituting and , we can integrate with respect to time.

The Substitution

The Key Formula

where , , and

We start with and substitute the parametric forms:

and

The limits change from -values to -values:

If decreases as increases, you're integrating right-to-left, which gives a negative area. You may need to negate the result or adjust limits.

Step 1: Find the Derivatives

Step 2: Use Symmetry

In the first quadrant (), x goes from 0 to 2 (left to right).

Area = 4 × (first quadrant):

Step 3: Simplify the Integral

Using :

Step 4: Integrate and Evaluate

Final Answer

✓ Matches the ellipse formula:

Step 1: Find dx/dt

Step 2: Set Up the Integral

Step 3: Expand the Square

Using :

Step 4: Integrate

Step 5: Evaluate at Bounds

At :

Final Answer

The area under one arch is exactly 3 times the area of the generating circle!