Welcome back to Professor Baker's Math Class! As we approach the end of Chapter 10, it is time to consolidate everything we have learned about Parametric Equations and Polar Coordinates. This review covers the essential calculus applications for these coordinate systems, including derivatives, tangents, arc length, and area.

Below, you will find a breakdown of the review test with key takeaways and formulas to help guide your study session.

1. Parametric Derivatives and Tangent Lines

The first step in mastering parametric equations is understanding how to find slopes and tangent lines. As seen in Problem 1 ($x = t^3 - 3t$, $y = t^2 - 3$), remember that the derivative is found using the chain rule:

$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

Key Concept: To find a horizontal tangent, set the numerator ($\frac{dy}{dt}$) to zero. To find a vertical tangent, set the denominator ($\frac{dx}{dt}$) to zero (provided the numerator isn't also zero).

2. Applications of Parametric Integration

Problems 2, 3, and 4 focus on the three major integral applications in parametric mode.

  • Area: Problem 2 asks for the area of an ellipse ($x = a\cos\theta, y = b\sin\theta$). The formula used is $A = \int y dx$ (or specifically $A = \int y(t) x'(t) dt$). Don't forget to use symmetry to simplify your bounds!
  • Arc Length: For Problem 3, we calculate the length of the curve using: $$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$
  • Surface Area of Revolution: Problem 4 involves rotating a curve around the x-axis. The formula combines the arc length element with the circumference: $$S = \int_a^b 2\pi y(t) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$

3. Polar Coordinates: Slopes and Area

Switching gears to polar coordinates, Problem 5 reminds us that finding the slope involves the product rule because $x = r\cos\theta$ and $y = r\sin\theta$. The formula can look intimidating, but if you derive it step-by-step from $x$ and $y$, you will be fine.

The Polar Area Formula:

Problem 6 ($r = 2 + \sin 4\theta$) requires finding the area enclosed by the curve. Remember the golden rule for polar area:

$$A = \int_{\alpha}^{\beta} \frac{1}{2} r^2 \, d\theta$$

Study Tip: When squaring $r$, you often encounter terms like $\sin^2(k\theta)$ or $\cos^2(k\theta)$. Be prepared to use the power-reduction identities:

$$\sin^2 u = \frac{1 - \cos(2u)}{2} \quad \text{and} \quad \cos^2 u = \frac{1 + \cos(2u)}{2}$$

4. Polar Arc Length

Finally, Problem 7 ($r = \theta^2$) tests the arc length in polar coordinates. The formula differs slightly from the parametric version:

$$L = \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$$

This review covers a lot of ground, but these formulas are the backbone of Chapter 10. Work through the attached PDF to see the handwritten solutions for every step of the algebra. Good luck studying!